Electronic Structure of Actinides A Dynamical Mean Field

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

Overview • DMFT in actinides and their compounds (Spectral density functional approach). Examples: – Plutonium, Americium, Curium. – Compounds: Pu. O 2, Pu. Am 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)

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).

DMFT + electronic structure method 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 Effective (DFT-like) single particle Spectrum consists of delta like peaks Spectral density usually contains renormalized quasiparticles and Hubbard bands

How good is single site DMFT for f systems? Elements: Compounds: f 5 L=5, S=5/2 J=5/2 Pu. O 2 Pu. Am f 7 L=0, S=7/2 J=7/2 f 6 L=3, S=3 J=0

Overview of actinides Many phases Two phases of Ce, a and g with 15% volume difference 25% increase in volume between a and d phase

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

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

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

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 ØNo magnetic moments, Ølarge mass ØLarge specific heat, ØMany phases, small or large volume One more electron in the f shell ØMagnetic moments! (Curie-Weiss law at high T, ØOrders antiferromagnetically at low T) ØSmall effective mass (small specific heat coefficient) ØLarge volume

Density functional based electronic structure calculations: ØAll Cm, Am, Pu are magnetic in LDA/GGA LDA: Pu(m~5 m. B), Am (m~6 m. B) Cm (m~4 m. B) Exp: Pu (m=0), Am (m=0) Cm (m~7. 9 m. B) ØNon magnetic LDA/GGA predicts volume up to 30% off. ØTreating f’s as core overestimates volume of d-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 m. B close to LS coupling) Plutonium dynamically restores symmetry -> becomes paramagnetic

Multiplet structure crucial for correct Tk in Pu (~800 K) and reasonable Tc in Cm (~100 K) Without F 2, F 4, F 6: 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) Pu partly f 5 partly f 6 F electron fluctuates between these atomic states on the time scale t~h/Tk (femtoseconds) One dominant atomic state – ground state of the atom

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

f-sumrule for core-valence conductivity Similar to optical conductivity: 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) B=B 0 - 4/15<l. s>/(14 -nf) B 0~3/5 Branching ratio depends on: • average SO coupling in the f-shell <l. s> • average number of holes in the f-shell nf B. T. Tole and G. van de Laan, PRA 38, 1943 (1988) 4 d 5/2 ->5 f 7/2 4 d 3/2 ->5 f 5/2 Core splitting~50 e. V Energy loss [e. V] A 5/2 area under the 5/2 peak

B=B 0 - 4/15<l. s>/(14 -nf) LDA+DMFT One measured quantity B, two unknowns Close to atom (IC regime) Itinerancy tends to decrease <l. s> [a] G. Van der Laan et al. , PRL 93, 97401 (2004). [b] G. Kalkowski et al. , PRB 35, 2667 (1987) [c] K. T. Moore et al. , PRB 73, 33109 (2006).

Pu dioxide d-Pu nf~5. 2 Pu. O 2 nf~4. 3 3 spd electrons + 1 f electron of Pu is taken by 2 oxygens-> f^4

Optical conductivity 2 p->5 f 5 f->5 f Pu: similar to heavy fermions (Kondo type conductivity) Scale is large MIR peak at 0. 5 e. V Pu. O 2: typical semiconductor with 2 e. V gap, charge transfer

Pu-Am mixture, 50%Pu, 50%Am Lattice expands for 20% -> Kondo collapse is expected 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 d phase stabilized by shift to mixed valence nf~5. 2 ->nf~5. 4 f 6: Shorikov, et al. , PRB 72, 024458 (2005); Shick et al, Europhys. Lett. 69, 588 (2005). Pourovskii et al. , Europhys. Lett. 74, 479 (2006). 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 Pu. O 2

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

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

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

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

Hubbard model self-energy on imaginary axis, 2 x 2 Low frequency very different Far from Mott transition coherent Close to Mott transition Very incoherent Optimal doping in the t-J model (d~0. 16) has largest low energy self-energy Very incoherent at optimal doping Optimal doping in the Hubbard model (d~0. 1) has largest low energy self-energy Very incoherent at optimal doping

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 (2001)

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) F. Carbone et. al, cond-mat/0605209

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

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 actinides and their compounds, single site LDA+DMFT gives the zero-th order picture • 2 D 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

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)

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 ~ 1/Jeff line: cluster DMFT (cond-mat 0601478), symbols: Bi 2212, F. Carbone et. al, cond-mat/0605209

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

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

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

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

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 approach describes well the transition 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)

Monotonically increasing J an SOC

Americium

Americium f 6 -> L=3, S=3, J=0 Mott Transition? "soft" phase f localized "hard" phase f bonding Density functional based electronic structure calculations: q Non magnetic LDA/GGA predicts volume 50% off. A. Lindbaum, S. Heathman, K. Litfin, and Y. Méresse, J. -C. Griveau, J. Rebizant, G. H. Lander, and G. Kotliar Rev. Lett. 097002 Phys. Rev. B 63, 214101 q Magnetic GGA(2001) corrects most of error in. Phys. volume but 94, gives m(2005) ~6 m. B (Soderlind et. al. , PRB 2000). q Experimentally, Am has non magnetic f 6 ground state with J=0 (7 F 0)

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)

Trends in Actinides alpa->delta volume collapse transition F 0=4, F 2=6. 1 F 0=4. 5, F 2=7. 15 Curie-Weiss Same transition in Am under pressure F 0=4. 5, F 2=8. 11 Curium has large magnetic moment and orders antif. Tc

LS versus jj coupling in Actinides • Occupations non-integer except Cm • Close to intermediate coupling • Delocalization in U & Pu-> towards LS • Am under pressure goes towards LS • Curium is localized, but close to LS! m=7. 9 m. B not m=4. 2 m. B 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, 85109 (2005) d a

Localization versus itinerancy in the periodic table?