Understanding f-electron materials using Dynamical Mean Field Theory Understanding f-electron materials using Dynamical Mean Field Theory Gabriel Kotliar and Center for Materials Theory $upport : NSF -DMR DOE-Basic Energy Sciences Collaborators: K. Haule and J. Shim Solid State Seminar U. Oregon January 15 th

Outline Introduction to Correlated Materials Introduction to Dynamical Mean Field Theory Applications to f electrons: CeIrIn5 URu2Si2 Pu-Am-Cm PuSe PuTe Conclusions

Excitation spectrum of a Fermi system has the same structure as the excitation spectrum of a perfect Fermi gas. Landau Fermi Liquid Excitation spectrum of a Fermi system has the same structure as the excitation spectrum of a perfect Fermi gas. Electrons in a Solid:the Standard Model Kohn Sham Density Functional Theory Rigid bands, optical transitions, thermodynamics, transport……… Static Mean Field Theory. 2 Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) n band index, e.g. s, p, d,,f Bloch waves in a periodic potential

Quantum mechanical description of the states in metals and semiconductors. Bloch waves. En(k). Inhomogenous systems. Doping. Theory of donors and acceptors. Interfaces. p-n junctions. Transistors. Integrated circuits computers. Physical Insights into Materials -> Technology GW= First order PT in screened Coulomb interactions around LDA 3

Correlated materials: simple recipe Transition metal oxides Oxygen transition metal ion Cage : e.g 6 oxygen atoms (octahedra) or other ligands/geometry Build a microscopic crystal with this building block Layer the structure Transition metal inside Transition metal ions Rare earth ions Actinides 4

Lix CoO 2 Na x CoO 2 Lix CoO 2 Na x CoO 2 YBa2Cu3O7 VO2 5

How do we know that the electrons are heavy ? Heavy Fermions: intermetallics containing 4f elements Cerium, and 5f elements Uranium. Broad spd bands + atomic f open shells.

Heavy Fermion Metals T(K) CeAl 3 UBe 13  -1 (emu/mol ) Coherence Incoherence Crossover Magnetic Oscillations

A Very Selected Class of HF

URu2Si2 U Si Ru A signature problem ?

Correlated Electron Systems Pose Basic Questions in CMT: from atoms to solids Correlated Electron Systems Pose Basic Questions in CMT: from atoms to solids How to describe electron from localized to itinerant ? How do the physical properties evolve ? Non perturbative techniques Needed!! (Dynamical) mean field theory for this problem, 8

Classical case Quantum case A. Georges, G. Kotliar (1992) Mean-Field : Classical vs Quantum Easy!!! Hard!!! but doable QMC, PT, ED, DMRG……. Prushke T. et. al Adv. Phys. (1995) Georges Kotliar Krauth Rosenberg RMP (1996) Kotliar et. al. RMP (2006),……………………………………...

Dynamical Mean Field Theory Describes the electron both in the itinerant (wave-like) and localized (particle-like) regimes and everything in between!. Follow different mean field states (phases) Compare free energies. Non Gaussian reference frame for correlated materials. Reference frame can be cluster of sites CDMFT 11

Determine energy and and  self consistently from extremizing a functional. Savrasov and Kotliar PRB 69, , (2001) Full self consistent implementation Determine energy and and  self consistently from extremizing a functional. Savrasov and Kotliar PRB 69, , (2001) Full self consistent implementation 12 Spectra=- Im G(k,  ) LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

DMFT Concepts DMFT Concepts Valence Histograms. Describes the history of the “atom” in the solid, multiplets! Weiss Weiss field, collective hybridizationfunction, quantifies the degree of localization Weiss Weiss field, collective hybridizationfunction, quantifies the degree of localization Functionals of density and spectra give total energies

Photoemission Spectral functions and the State of the Electron Photoemission Spectral functions and the State of the Electron Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2  e Angle integrated spectra 9 a)Weak correlations b)Strong correlation: FL parameters can’t be evaluated in PT or FLT does not work. A(k, 

Qualitative Phase diagram :frustrated Hubbard model, integer filling M. Rozenberg et.al. PRL,75, 105 (1995) T/W 13 CONCEPT: (orbitally resolved) spectral function. Transfer of spectral weight. CONCEPT: (orbital selective) Mott transition. CONCEPT: Quasiparticle bands, T*, and Hubbard bands

Outline Introduction to Correlated Materials Introduction to Dynamical Mean Field Theory Applications to f electrons : CeIrIn5 Pu-Am-Cm PuSe PuTe URu2Si2 Conclusions

 CeRhIn5: TN=3.8 K;   450 mJ/molK2  CeCoIn5: Tc=2.3 K;   1000 mJ/molK2;  CeIrIn5: Tc=0.4 K;   750 mJ/molK2 4f’s heavy fermions, 115’s, CeMIn 5 M=Co, Ir, Rh out of plane in-plane Ce In Ir 21

At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV) At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV) At 10K: At 10K: very narrow Drude peak very narrow Drude peak First MI peak at 0.03eV~250cm -1 First MI peak at 0.03eV~250cm -1 Second MI peak at 0.07eV~600cm -1 Second MI peak at 0.07eV~600cm -1 Optical conductivity in LDA+DMFT Shim, HK Gotliar Science (2007) ‏ K. Burch et.al. D. Basov et.al.

Ce In In Structure Property Relation: Ce115’s Optics and Multiple hybridization gaps 300K eV10K Larger gap due to hybridization with out of plane InLarger gap due to hybridization with out of plane In Smaller gap due to hybridization with in- plane InSmaller gap due to hybridization with in- plane In non-f spectra J. Shim et. al. Science

Difference between Co,Rh,Ir 115’s more localized more itinerant IrCoRh superconducting magnetically ordered “ good ” Fermi liquid Total and f DOS f DOS Haule Yee and Kim arXiv: Haule Yee and Kim arXiv:

URu2Si2 U Si Ru A signature problem ?

Two Broken Symmetry Solutions Hidden Order LMA K. Haule and GK Weiss field

Order parameter: Different orientation gives different phases: “adiabatic continuity” explained. Hexadecapole order testable by resonant X-ray In the atomic limit: Hidden order parameter Paramagnetic phase low lying singlets f^2 Valence Histogram

Mean field Exp. by E. Hassinger et.al. PRL 77, (2008) Simplified toy model phase diagram mean field theory

Orbitally resolved DOS

DMFT “STM” URu2Si2 T=20 K Fano lineshape: q~1.24,  ~6.8meV, very similar to exp Davis U Si Ru Si

Lattice response

Localization Delocalization in Actinides Mott Transition  Modern understanding of this phenomenaDMFT.  Pu  17

Total Energy as a function of volume for Pu (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Moment is first reduced by orbital spin moment compensation. The remaining moment is screened by the spd and f electrons The f electron in  - phase is only slightly more localized than in the  -phase which has larger spectral weight in the quasiparticle peak and smaller weight in the Hubbard bands

Localization Delocalization in Actinides Mott Transition  Modern understanding of this phenomenaDMFT.  Pu  17

The standard model of solids fails near Pu Spin Density functional theory: Pu, Am, magnetic, large orbital and spin moments. Experiments (Lashley et. al. 2005, Heffner et al. (2006)):  Pu is non magnetic. No static or fluctuating moments. Susceptibility, specific heat in a field, neutron quasielastic and inelastic scattering, muon spin resonance… Paramagnetic LDA underestimates Volume of  Pu. Paramagnetic LDA underestimates Volume of  Pu. Thermodynamic and transport properties similar to strongly correlated materials. Thermodynamic and transport properties similar to strongly correlated materials. Plutonium: correlated paramagnetic metal. Plutonium: correlated paramagnetic metal.

DMFT Phonons in fcc  -Pu ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

K.Haule J. Shim and GK Nature 446, 513 (2007) Trends in Actinides alpa->delta volume collapse transition Curium has large magnetic moment and orders antiferromagnetically Pu does is non magnetic. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

Photoemission Photoemission Havela et. al. Phys. Rev. B 68, (2003) Havela et. al. Phys. Rev. B 68, (2003)

What is the valence in the late actinides ? Plutonium has an unusual form of MIXED VALENCE

LDA results Finding the f occupancyTobin et. al. PRB 72, K. Moore and G. VanDerLaan RMP (2009). Shim et. al. Europhysics Lett (2009) DMFT results

Localization delocalization of f electrons in compounds. Pu Chalcogenides [PuSe, PuS, PuTe]: Pauli susceptibility, small gap in transport. Pu Pnictides [PuP, PuAs, PuSb], order magnetically. Simple cubic NaCl structure Going from pnictides to chalcogenides tunes the degree of localization of the f electron. Earlier work Shick et. al. Pourovski et. al.

LDA+DMFT C. Yee Expts. T. Rurakiewicz et. al. PRB 70,

PuTe: a 5f mixed valent semi- conductor PuSb: a local moment metal

Summary Correlated Electron Systems. Huge Phase Space. Fundamental questions. Promising applications. DMFT reference frame to think about electrons in solids. Quasiparticles Hubbard bands. Compare with the standard model. Many succesful applications, some examples illustrating a) the concepts, b) the role of realistic modelling, and c) the connection between theory and experiment and the role of theoretical spectroscopy. 28

Conclusion: DMFT provides a surprisingly accurate description of f electron systems. It’s physical content at very low temperatures is that of a heavy Fermi liquid in common with other methods but asymptotia is hardly reached (and relevant). Complete description of the crossover. Variety and Universality.

Outlook “Locality “ as an alternative to Perturbation Theory. Needed: progress in implementation. e.g. full solution of DMFT equations on a plaquette, robust GW+DMFT …………. Fluctuation around DMFT. Interfaces, junctions, heterostructures……….. Motterials, Materials,……. Towards rational material design with correlated electrons systems 28

Looking for moments. Pu under (negative ) pressure. C Marianetti, K Haule GK and M. Fluss Phys. Rev. Lett. 101, (2008)

Conclusion: some general comments. DMFT approach. Can now start from the material. Can start from high energies, high temperatures, where the method (I believe ) is essentially exact, far from critical points, provided that one starts from the right “reference frame”. Spectral “fingerprints” and their chemical origin. Still need better tools to analyze and solve the DMFT equations. Still need simpler approaches to rationalize simpler limit. Validates some aspects of slave boson mean field theories, modifies quantitatively and sometimes qualitatively the answers.

At lower temperatures, one has to study different broken symmetry states. At lower temperatures, one has to study different broken symmetry states. Compare free energies, draw phase diagram Beyond DMFT: Write effective low energy theories that match the different regions of the phase diagram. Close contact with experiments. Many materials are being tried, methods are being refined Contemplating material design using correlated electron systems.

Very slow crossover! T*T*T*T* Buildup of coherence in single impurity case TKTKTKTK coherent spectral weight T scattering rate coherence peak Buildup of coherence Crossover around 50K Slow crossover compared to AIM

Plutonium