Electronic Structure of Strongly Correlated Materials:a Dynamical Mean Field Theory (DMFT) approach Gabriel Kotliar Physics Department and Center for Materials.

Electronic Structure of Strongly Correlated Materials:a Dynamical Mean Field Theory (DMFT) approach Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University University of Washington Seattle May 10 th 2005

Outline Introduction to strongly correlated electrons and Dynamical Mean Field Theory (DMFT). The Mott transition problem. Theory and experiments. More realistic calculations. Pu the Mott transition across the actinide series. Conclusions. Current developments and future directions.

C. Urano et. al. PRL 85, 1052 (2000) Strong Correlation Anomalies cannot be understood within the standard model of solids, based on a RIGID BAND PICTURE,e.g.“Metallic “resistivities that rise without sign of saturation beyond the Mott limit, temperature dependence of the integrated optical weight up to high frequency

Two paths for calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

Model Hamiltonians: Hubbard model  U/t  Doping  or chemical potential  Frustration (t’/t)  T temperature

Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

Mean-Field Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992) Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

DMFT as an approximation to the Baym Kadanoff functional

Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)Phys. Rev. B 69, 205108 (2004)

U/t=4. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

One Particle Spectral Function and Angle Integrated Photoemission Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 Probability of absorbing an electron and transfering energy  =Ei-Ef, and momentum k (1-f(  )) A(  ) M 2 Theory. Compute one particle greens function and use spectral function. e e

Photoemission and the Theory of Electronic Structure Limiting case itinerant electrons Limiting case localized electrons Hubbard bands Local Spectral Function

Pressure Driven Mott transition How does the electron go from the localized to the itinerant limit ?

T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT. High temperature universality M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

V2O3:Anomalous transfer of spectral weight Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett., 21 (1993), 593 M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

Anomalous Resistivity and Mott transition Ni Se 2-x S x Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

Single site DMFT and kappa organics

Ising critical endpoint! In V 2 O 3 P. Limelette et.al. Science 302, 89 (2003)

. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, 186403 (2003).

Conclusions. Three peak structure, quasiparticles and Hubbard bands. Non local transfer of spectral weight. Large metallic resistivities. The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. Coherent and incoherence crossover. Real and momentum space. Theory and experiments begin to agree on a broad picture.

Two paths for calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

The combination of realistic band theory and many body physics, is a very broad subject. Having a practical and tractable non perturbative method for solving many body Hamiltonians, the next step is to bring more realistic descriptions of the materials Orbital degeneracy and realistic band structure. LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). The light, sp (or spd) electrons are extended, well described by LDA.The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting average energy already contained in LDA. Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, from DMFT. Determine the density self consistently.(Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, Abrahams, Nature 2001).

LDA+DMFT Self-Consistency loop DMFT U Edc

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001). Phys. Rev. B 62, 12715 (2000) Ex. Ir>=|R,  > Gloc=G(R , R  ’)  R,R’ ’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. Sum of 2PI graphs One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.

Next Step: GW+EDMFT S. Savrasov and GK.(2001). P.Sun and GK. (2002). S. Biermann F. Aersetiwan and A.Georges. (2002). P Sun and G.K (2003) Implementation in the context of a model Hamiltonian with short range interactions.P Sun and G. Kotliar cond-matt 0312303 or with a static U on heavy electrons, without self consistency. Biermann et.al. PRL 90,086402 (2003) W W

Actinies, role of Pu in the periodic table

Pu phases: A. Lawson Los Alamos Science 26, (2000) LDA underestimates the volume of fcc Pu by 30%. Within LDA fcc Pu has a negative shear modulus. LSDA predicts  Pu to be magnetic with a 5  b moment. Experimentally it is not. Treating f electrons as core overestimates the volume by 30 %

Small amounts of Ga stabilize the  phase (A. Lawson LANL)

Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

Total Energy as a function of volume for Pu Total Energy as a function of volume for Pu W (ev) vs (a.u. 27.2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2004)

Double well structure and  Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Phys vol.126, 1009 27. (2002)]See also A. Lawson et.al.Phil. Mag. B 82, 1837 ]

Phonon Spectra Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured.

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109,2003

K. Haule, Pu- photoemission with DMFT using vertex corrected NCA.

Dynamical Mean Field View of Pu ( Savrasov Kotliar and Abrahams, Nature 2001) 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 earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

Pu strongly correlated element, at the brink of a Mott instability. Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu. Clues to understanding other Pu anomalies.

Outline Introduction to strongly correlated electrons. Introduction to Dynamical Mean Field Theory (DMFT) The Mott transition problem. Theory and experiments. More realistic calculations. Pu the Mott transition across the actinide series. Conclusions. Current developments and future directions.

Conclusion DMFT. Electronic Structure Method under development. Local Approach. Cluster extensions. Quantitative results, connection between electronic structure, scales and bonding. Qualitative understanding by linking real materials to impurity models. Concepts to think about correlated materials. Closely tied to experiments. System specific. Many materials to be studied, realistic matrix elements for each spectroscopy. Optics.……

Some References Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, (1996). Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

DMFT : What is the dominant atomic configuration,what is the fate of the atomic moment ? Snapshots of the f electron :Dominant configuration:(5f) 5 Naïve view Lz=-3,-2,-1,0,1, ML=-5  B,, S=5/2 Ms=5  B. Mtot=0 More realistic calculations, (GGA+U),itineracy, crystal fields     ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000) This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce).  Contrast Am:(5f) 6

Anomalous Resistivity PRL 91,061401 (2003)

The delta –epsilon transition The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

Epsilon Plutonium.

Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Further Approximations. o The light, SP (or SPD) electrons are extended, well described by LDA.The heavy, d(or f) electrons are localized treat by DMFT.LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term). o Truncate the W operator act on the H sector only. i.e. Replace W(  ) by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g. M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

or the U matrix can be adjusted empirically. At this point, the approximation can be derived from a functional (Savrasov and Kotliar 2001) FURTHER APPROXIMATION, ignore charge self consistency, namely set LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also. A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988). Reviews: Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65. Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p. 428. Georges, A., 2004, Electronic Archive,.lanl.gov, condmat/ 0403123.

Realistic DMFT loop

LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

Mott transition into an open (right) and closed (left) shell systems. AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ???

Americium under pressure 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 ) Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition?“Soft” “Hard”

Mott transition in open (right) and closed (left) shell systems. S S U U  T Log[2J+1] Uc  ~1/(Uc-U) J=0 ??? Tc

Am under pressure: J.C. GriveauJ. Rebizant G. Lander and G. Kotliar PRL (2005)

J. C. Griveau et. al. (2004)

Am Equation of State: LDA+DMFT Predictions LDA+DMFT predictions:  Non magnetic f 6 ground state with J=0 ( 7 F 0 )  Equilibrium Volume: V theory /V exp =0.93  Bulk Modulus: B theory =47 GPa Experimentally B=40-45 GPa Theoretical P(V) using LDA+DMFT Self-consistent evaluations of total energies with LDA+DMFT using matrix Hubbard I method. Accounting for full atomic multiplet structure using Slater integrals: F (0) =4.5 eV, F (2) =8 eV, F (4) =5.4 eV, F (6) =4 eV New algorithms allow studies of complex structures. Predictions for Am II Predictions for Am IV Predictions for Am III Predictions for Am I

Photoemission Spectrum from 7 F 0 Americium LDA+DMFT Density of States Experimental Photoemission Spectrum (after J. Naegele et.al, PRL 1984) Matrix Hubbard I Method F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV

Atomic Multiplets in Americium LDA+DMFT Density of States Exact Diag. for atomic shell F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV Matrix Hubbard I Method F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV

Anomalous Resistivity PRL 91,061401 (2003)

The Mott Transiton across the Actinides Series.

Pu phases: A. Lawson Los Alamos Science 26, (2000) LDA underestimates the volume of fcc Pu by 30%. Within LDA fcc Pu has a negative shear modulus. LSDA predicts  Pu to be magnetic with a 5  b moment. Experimentally it is not. Treating f electrons as core overestimates the volume by 30 %

Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

Double well structure and  Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Phys vol.126, 1009 27. (2002)]See also A. Lawson et.al.Phil. Mag. B 82, 1837 ] Double well structure is immediate consequence of having two solutions to the DMFT equations and allowing the relaxation of the volume. The itinerant solution expands the localized solution contracts.