First principle calculations of plutonium and plutonium compounds: part 1 A. B. Shick Institute of Physics ASCR, Prague, CZ
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 2016 2
"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, 1832 15/09/16 3
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
Electronic Structure Theory Many-Body Interacting Problem
Density Functional Theory
20 th Century approach
Kohn-Sham Equations
Ab-initio calculations Local/Semi-Local Density Approximation (LDA/GGA)
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 ) 5 4 3 2 1 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. 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
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
LDA vs LDA+U: NiO Band Gap (ev): LSDA LSDA+U Experiment 0.4 3.4 4.0
LDA vs LDA+U: Gd LSDA: Anti-FM GS Exp.: Ferro-M GS LSDA+U: Ferro-M GS
...continued PRB 50, 16861 (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
...continued PRB 50, 16861 (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 1 2 1 2 n 1 2 n 11 2 2 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
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. 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. 19 15 September 2016
EXTRA SLIDES
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
[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
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
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
Kohn-Sham Dirac Eqs. Scalar-relativistic Eqs. SOC -