Collaborators: G.Kotliar, Ji-Hoon Shim, S. Savrasov Kristjan Haule, Physics Department and Center for Materials Theory Rutgers University Electronic Structure of Actinides: A Dynamical Mean Field Perspective. UC Davis

DMFT in actinides and their compounds (Spectral density functional approach). Examples: –Plutonium, Americium, Curium. –Compounds: PuO 2, PuAm Observables: –Valence, Photoemission, and Optics, X-ray absorption Extensions of DMFT to clusters. Examples: –Coherence in the Hubbard and t-J model New general impurity solver (continuous time QMC) developed (can treat clusters and multiplets) Overview

V2O3V2O3 Ni 2-x Se x  organics Universality of the Mott transition First order MIT Critical point Crossover: bad insulator to bad metal 1B HB model (DMFT):

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

DMFT + electronic structure method Effective (DFT-like) single particle Spectrum consists of delta like peaks Spectral density usually contains renormalized quasiparticles and Hubbard bands Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites problem, in a medium of non interacting electrons obeying a self-consistency condition. (A. Georges et al., RMP 68, 13 (1996)). DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated bands (s,p): use LDA or GW For correlated bands (f or d): with DMFT add all local diagrams

How good is single site DMFT for f systems? f5 L=5,S=5/2 J=5/2 f6 L=3,S=3 J=0 f7 L=0,S=7/2 J=7/2 PuO2 PuAm Compounds: Elements:

Overview of actinides Two phases of Ce,  and  with 15% volume difference 25% increase in volume between  and  phase Many phases

Trivalent metals with nonbonding f shell f’s participate in bonding Partly localized, partly delocalized Overview of actinides?

Why is Plutonium so special? Heavy-fermion behavior in an element No curie Weiss up to 600K Typical heavy fermions (large mass->small Tk Curie Weis at T>Tk)

Ga doping stabilizes  -Pu at low T, lattice expansion Am doping -> lattice expansion Expecting unscreened moments! Does not happen! Plutonium puzzle?

Curium versus Plutonium nf=6 -> J=0 closed shell (j-j: 6 e- in 5/2 shell) (LS: L=3,S=3,J=0) One hole in the f shell One more electron in the f shell  No magnetic moments,  large mass  Large specific heat,  Many phases, small or large volume  Magnetic moments! (Curie-Weiss law at high T,  Orders antiferromagnetically at low T)  Small effective mass (small specific heat coefficient)  Large volume

Density functional based electronic structure calculations:  All Cm, Am, Pu are magnetic in LDA/GGA LDA: Pu(m~5   ), Am (m~6   ) Cm (m~4   ) Exp: Pu (m=0), Am (m=0) Cm (m~7.9   )  Non magnetic LDA/GGA predicts volume up to 30% off.  Treating f’s as core overestimates volume of  -Pu, reasonable volume for Cm and Am Can LDA+DMFT predict which material is magnetic and which is not?

Starting from magnetic solution, Curium develops antiferromagnetic long range order below Tc above Tc has large moment (~7.9  close to LS coupling) Plutonium dynamically restores symmetry -> becomes paramagnetic

Multiplet structure crucial for correct Tk in Pu (~800K) and reasonable Tc in Cm (~100K) Without F2,F4,F6: Curium comes out paramagnetic heavy fermion Plutonium weakly correlated metal

Valence histograms Density matrix projected to the atomic eigenstates of the f-shell (Probability for atomic configurations) F electron fluctuates between these atomic states on the time scale t~h/Tk (femtoseconds) One dominant atomic state – ground state of the atom Pu partly f 5 partly f 6

core valence 4d 3/2 4d 5/2 5f 5/2 5f 7/2 Excitations from 4d core to 5f valence Electron energy loss spectroscopy (EELS) or X-ray absorption spectroscopy (XAS) Energy loss [eV] Core splitting~50eV 4d 5/2 ->5f 7/2 4d 3/2 ->5f 5/2 Measures unoccupied valence 5f states Probes high energy Hubbard bands! hv Core splitting~50eV Probe for Valence and Multiplet structure: EELS&XAS A plot of the X-ray absorption as a function of energy

Current: Expressed in core valence orbitals: The f-sumrule: can be expressed as Branching ration B=A 5/2 /(A 5/2 +A 3/2 ) Energy loss [eV] Core splitting~50eV 4d 5/2 ->5f 7/2 4d 3/2 ->5f 5/2 B=B0 - 4/15 /(14-n f ) A 5/2 area under the 5/2 peak Branching ratio depends on: average SO coupling in the f-shell average number of holes in the f-shell n f B 0 ~3/5 B.T. Tole and G. van de Laan, PRA 38, 1943 (1988) Similar to optical conductivity: f-sumrule for core-valence conductivity

One measured quantity B, two unknowns Close to atom (IC regime) Itinerancy tends to decrease B=B0 - 4/15 /(14-n f ) [a] G. Van der Laan et al., PRL 93, (2004). [b] G. Kalkowski et al., PRB 35, 2667 (1987) [c] K.T. Moore et al., PRB 73, (2006). LDA+DMFT

 -Pu nf~5.2 PuO 2 nf~4.3 3spd electrons + 1f electron of Pu is taken by 2 oxygens-> f^4 Pu dioxide

2p->5f 5f->5f Pu: similar to heavy fermions (Kondo type conductivity) Scale is large MIR peak at 0.5eV PuO 2 : typical semiconductor with 2eV gap, charge transfer Optical conductivity

Pu-Am mixture, 50%Pu,50%Am Lattice expands for 20% -> Kondo collapse is expected f 6 : Shorikov, et al., PRB 72, (2005); Shick et al, Europhys. Lett. 69, 588 (2005). Pourovskii et al., Europhys. Lett. 74, 479 (2006). Could Pu be close to f 6 like Am? Inert shell can not account for large cv anomaly Large resistivity! Our calculations suggest charge transfer Pu  phase stabilized by shift to mixed valence nf~5.2->nf~5.4 Hybridization decreases, but nf increases, Tk does not change significantly!

What is captured by single site DMFT? Captures volume collapse transition (first order Mott-like transition) Predicts well photoemission spectra, optics spectra, total energy at the Mott boundary Antiferromagnetic ordering of magnetic moments, magnetism at finite temperature Branching ratios in XAS experiments, Dynamic valence fluctuations, Specific heat Gap in charge transfer insulators like PuO2

Beyond single site DMFT What is missing in DMFT? Momentum dependence of the self-energy m*/m=1/Z Various orders: d-waveSC,… Variation of Z, m*,  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

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

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

New continuous time QMC, expansion in terms of hybridization General impurity problem Diagrammatic expansion in terms of hybridization  +Metropolis sampling over the diagrams Contains all: “Non-crossing” and all crossing diagrams! No problem with multiplets k

Hubbard model self-energy on imaginary axis, 2x2 Far from Mott transition coherent Low frequency very different Close to Mott transition Very incoherent Optimal doping in the t-J model  ~0.16) has largest low energy self-energy Very incoherent at optimal doping Optimal doping in the Hubbard model (  ~0.1) has largest low energy self-energy Very incoherent at optimal doping

Insights into superconducting state (BCS/non-BCS)? J. E. Hirsch, Science, 295, 5563 (2001) 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)

Bi2212 ~1eV Weight bigger in SC, K decreases (non-BCS) Weight smaller in SC, K increases (BCS-like) Optical weight, plasma frequency F. Carbone et.al, cond-mat/

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

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

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

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 actinides and their compounds, single site LDA+DMFT gives the zero-th order picture 2D models of high-Tc require cluster of sites. Some aspects of optimally doped regime can be described with cluster DMFT on plaquette: –Large scattering rate in normal state close to optimal doping –Evolution from kinetic energy saving to BCS kinetic energy cost mechanism Conclusions

Pengcheng et.al., Science 284, (1999) YBa 2 Cu 3 O 6.6 (Tc=62.7K) 41meV resonance local susceptibility Resonance at 0.16t~48meV Most pronounced at optimal doping Second peak shifts with doping (at 0.38~120meV opt.d.) and changes below Tc – contribution to condensation energy

Optics mass and plasma frequency Extended Drude model In sigle site DMFT plasma frequency vanishes as 1/Z (Drude shrinks as Kondo peak shrinks) at small doping Plasma frequency vanishes because the active (coherent) part of the Fermi surface shrinks In cluster-DMFT optics mass constant at low doping doping ~ 1/Jeff line: cluster DMFT (cond-mat ), symbols: Bi2212, F. Carbone et.al, cond-mat/

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

Partial DOS 4f 5d 6s Z=0.33

Pseudoparticle insight N=4,S=0,K=0 N=4,S=1,K=(  ) N=3,S=1/2,K=( ,0) N=2,S=0,K=0 A(  )  ’’ (  ) PH symmetry, Large 

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 spacecluster in real space

Cerium

Ce overview volumesexp.LDALDA+U  28Å Å 3  34.4Å Å 3 Transition is 1.order ends with CP  isostructural phase transition ends in a critical point at (T=600K, P=2GPa)   (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 

LDA and LDA+U f DOS total DOS volumesexp.LDALDA+U  28Å Å 3  34.4Å Å 3 ferromagnetic

LDA+DMFT alpha DOS T K (exp)= K

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

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

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

Monotonically increasing J an SOC

Americium

"soft" phase f localized "hard" phase f bonding Mott Transition? f 6 -> L=3, S=3, J=0 A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse, Phys. Rev. B 63, (2001) J.-C. Griveau, J. Rebizant, G. H. Lander, and G.Kotliar Phys. Rev. Lett. 94, (2005) Density functional based electronic structure calculations:  Non magnetic LDA/GGA predicts volume 50% off.  Magnetic GGA corrects most of error in volume but gives m ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 ( 7 F 0 )

Am within LDA+DMFT S. Y. Savrasov, K. Haule, and G. Kotliar Phys. Rev. Lett. 96, (2006) F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV Large multiple effects:

Am within LDA+DMFT n f =6 Comparisson with experiment from J=0 to J=7/2 “Soft” phase very different from  Ce not in local moment regime since J=0 (no entropy) "Hard" phase similar to  Ce, Kondo physics due to hybridization, however, nf still far from Kondo regime n f =6.2 Different from Sm! Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. Müller Phys. Rev. Lett. 52, (1984) Theory: S. Y. Savrasov, K. Haule, and G. Kotliar Phys. Rev. Lett. 96, (2006) V=V 0 Am I V=0.76V 0 Am III V=0.63V 0 Am IV

Curie-Weiss Tc Trends in Actinides alpa->delta volume collapse transition Same transition in Am under pressure Curium has large magnetic moment and orders antif. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

LS versus jj coupling in Actinides K.T.Moore, et.al.,PRB in press, 2006 G. Van der Laan, et.al, PRL 93,27401 (2004) J.G. Tobin, et.al, PRB 72, (2005) Occupations non-integer except Cm Close to intermediate coupling Am under pressure goes towards LS Delocalization in U & Pu-> towards LS   Curium is localized, but close to LS!  =7.9  B not  =4.2  B

Localization versus itinerancy in the periodic table?