First principle calculations of plutonium and plutonium compounds: part 1

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1 First principle calculations of plutonium and plutonium compounds: part 1 A. B. Shick Institute of Physics ASCR, Prague, CZ

2 Outline: u Lecture 1: Methods of Correlated band theory DFT and DFT+U u Lecture 2: Electronic and magnetic character of Pu DFT+Exact Diagonalization of the Anderson Model u Lecture 3: PuCoGa5 and related compounds Racah Materials 15 September

3 "The modern age has a false sense of security because of the great mass of data at its disposal. But the valid issue the extend to which people know how to form and master the material at their disposal." Goethe, /09/16 3

Crystal Structure Ø Materials properties on the electronic

4 Computational Materials Theory Periodic Table Ø Complex multiscale problem across a spectrum of length and time Crystal Structure Ø Materials properties on the electronic structure atomistic, and continuum levels Ø Tips for new materials and structures

5 Electronic Structure Theory Many-Body Interacting Problem

6 Density Functional Theory

7 20 th Century approach

8 Kohn-Sham Equations

9 Ab-initio calculations Local/Semi-Local Density Approximation (LDA/GGA)

Band picture: atomic orbitals overlap and form energy bands.

for magnetic prope E E E f Spin Split E f Favoring exchange over kinetic energy g(e)

Stoner model: Favorable to spend KE for a gain in exchange energy (Ig(E f ) > 1) g(e)

10 Band picture: atomic orbitals overlap and form energy bands. Delocalized arrow electrons bands with large become DOS atresponsible the Fermi Energy. for magnetic prope E E E f Spin Split E f Favoring exchange over kinetic energy g(e) g(e) Delocalized electrons become responsible for magnetic properties. Stoner model: Favorable to spend KE for a gain in exchange energy (Ig(E f ) > 1) g(e) Spin-up and -down bands spontaneously spin split. g(e) 6 Magnetic moment ( B ) Atom Bulk 0 Sc Ti V Cr Mn Fe Co Ni Cu

DFT+Spin-Orbit Coupling (LDA/GGA) Material N holes m tot s m d s m sp s Fe Co Ni 3.4 2.5 1.5 2.19 2.26-0.07 1.57 1.64 0.62 0.64-0.07-0.02 m orb 0.09 0.14 0.07 Data from O. Eriksson et al., Phys. Rev.

11 DFT+Spin-Orbit Coupling (LDA/GGA) Material N holes m tot s m d s m sp s Fe Co Ni m orb Data from O. Eriksson et al., Phys. Rev. B 42, 2707 (1990). Naturally incorporates spin and fine structure ( ) ( ) ( ) α = ( i h Ψ t = (cα (p qa c ) + βmc2 + qv )Ψ 0 σ σ 0 ) and β = ( I 0 0 I ), Ψ = ( Φ χ ) Leads to two coupled equations:two positive and two negative energy soluti p 2 ( 1 ( ) p 4 } 2m + {{ V (r) mc 2 + } 8 mc) 1 1dV 2m }{{} 2 c 2 (S L) h2 dv } r{{ dr } 4m 2 c 2 dr r }{{} Φ = EΦ Schrödinger mass correction spin orbit Darwin

12 Localized electrons Hund's rules: 1) Total spin S = i s i is maximized 2) Total orbital moment L = i l i is maximized 3) L and S couple parallel (J= L+S ) if el. shell is more than half-filled, L and S couple antiparallel (J= L-S ) if el. shell is less than half-filled

13 LDA vs LDA+U: NiO Band Gap (ev): LSDA LSDA+U Experiment

14 LDA vs LDA+U: Gd LSDA: Anti-FM GS Exp.: Ferro-M GS LSDA+U: Ferro-M GS

16 ...continued PRB 50, (94) LDA+U LDA correction E LDA+U = E LDA (ρ) + E corr (n i ) where, E corr = U 2 i =j n in j U 2 n l(n l 1) for N n l N +1and x = n l N E H = 1 UN(N 1) + UNx 2 E corr = 1 Ux(x 1) 2

17 ...continued PRB 50, (94) V corr = U( 1 2 x) 1-electron potential E corr / x: 1-electron potential E corr / x: integer occupation (x =0, 1) E corr vanishes but V co ergoes a jump of U. V corr = U( 1 2 x) V corr = U( 1 2 x) For integer occupation (x =0, 1) E corr vanishes but V corr For integer occupation (x =0, 1) E corr vanishes but V corr undergoes a jump of U. undergoes a jump of U. E V E V n 1 2 n Redifinition of the 1-particle energies: ϵ LDA+U Redifinition of the 1-particle i energies: ϵ LDA+U (N) =ϵ i i LDA (N) ± i U 2 U 2 depending on occupied (-) or empty (+) states. i (N) ± U 2 ending on occupied (-) or empty (+) states. LDA+U p.1 LDA+U

18 Mott transition in 5f-shell B. Johansson, 1975 Volume collapse M. Katsnelson et al., 1992

u Understanding of the physical and chemical properties of actinide materials. u 5f-electron challenge: Conventional band theory DFT (LDA/GGA) does not work well for heavy actinides.

19 u Understanding of the physical and chemical properties of actinide materials. u 5f-electron challenge: Conventional band theory DFT (LDA/GGA) does not work well for heavy actinides. u Beyond DFT: correlated band theory DFT+Orbital Polarization DFT+Self-Interaction Correction Hybrid Functionals (LDA/GGA + Hartree-Fock) DFT+Coulomb U DFT+Dynamical Mean Field Theory u Application to elemental δ-pu September 2016

20 EXTRA SLIDES

21 A.I. Lichtenstein & M.I. Katsnelson, PRB 57, 6884 (1998) N(E) Atomic physics (Coulomb-U) N(E) LDA d n SL> E F d n+ 1 E E F E N(E) LDA++ LHB QP UHB E F E

22 [G at (z)] 1 2 = 1 Z,µ µ c 1 c 2 µ Atomic Green z + Function E µ E + Eq.(5) µ H and exp (E Self-Energy µh N ) + exp Eq.(6) (E µ µ H N µ ) SIAM Green Function and local DM new n f f-occupation Construct LDA+U potential and solve K-S Eqs self-consistently Obtain on-site Green Function

resonance by STM Acknowledge collaboration with J. Kolorenc, A.

23 u LDA+ED calculations show the singlet many-body ground state for Co-in-bulk-Cu and Co-adatom on Cu 2 N u This is consitent with experimental observation of the Kondo resonance by STM Acknowledge collaboration with J. Kolorenc, A. Kozub (PhD student) Prague; A. I. Lichtenstein, Univestity of Hamburg; M. Etzkorn and H. Brune, EPFL, Lausanne, Switzerland

24 Projection to LAPW-basis G(z) = 1 V BZ BZ dk b m b b m z + µ b (k) b k(r) = G c b k+g k+g(r) k+g(r) = l,m [a lm k+gu l (r i )+b lm k+g u l (r i )]Y lm (ˆr i ) m b b m = u l Y lm b b u l Y lm + 1 < u u > u ly lm b b u l Y lm

25 Kohn-Sham Dirac Eqs. Scalar-relativistic Eqs. SOC -

Introduction to Density Functional Theory

1 Introduction to Density Functional Theory 21 February 2011; V172 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT 2 3 4 Ab initio Computational