Ab initio description of actinides: Hund’s exchange, spin-orbit coupling and crystal structure Bernard Amadon CEA, DAM, DIF , Arpajon, France www.cea.fr Conference: What about U ?
Actinides in the periodic table At atmospheric pressure: 4 d element: filling of the 4 d band (Bonding states and antibonding): 4 d electrons are delocalized. Lanthanides: 4 f electrons are localized , negligible overlap between 4 f orbitals . Actinide: intermediate case of localization. [Mac Mahan, et al J. Comp.-Aid. Mater. Des. 5, 131 (1998)] — Conference: What about U ? — 2/38
Plutonium has a large number of difference phases. Many phases Some phases with delocalized electrons (low volume) and phases with localized electrons (large volume). Pressure (kbar) 600 2 ) C ( ° e 4 r u t a 400 r 6 e p m 8 e T 200 1.260 δ δ′ ε 1.08 1.191 γ 1.125 1.06 e ζ m u 1.061 l Length 1.04 o V β 1.000 1.02 Pressure Instability . 1.00 6 0 0 0 α 4 2 0 ) 0 C ( ° Pressure (kbar) 4 e r u 2 a t 0 6 r 0 e p m 8 e T 1 0 0 From Kevin T. Moore and Gerrit van der Laan Rev. Mod. Phys. 81, 235 (2009) Figure 16. Plutonium Instability with Temperature and Pressure — Conference: What about U ? — 3/38 Plutonium is notoriously unstable under almost any external disturbance. Over a span of only 600°, it exhibits six different allotropic phases with large accompanying volume changes before it melts. Pressures on the order of kilobar (100 megapascals) are suffi- cient to squeeze out the high-volume allotropes (Morgan 1970). Small chemical additions can stabilize these high-volume phases. (Reproduced with permission from The Metallurgical Society.) Los Alamos Science
α -U, α -Np, and α -Pu are low symmetry phases induced by chemical bonding of f electrons. δ -Pu, Am, and Cm have compact phases (as lanthanides). δ and α Plutonium, have no local moment, ordered or not. (J. C. Lashley, A. Lawson, R. J. McQueeney, and G. H. Lander, Phys. Rev. B 72, 054416 (2005)) — Conference: What about U ? — 4/38
Non magnetic GGA underestimates volume for late actinides 32 Experiment 96 96 GGA (NM) 95,5 95,5 28 3 ) Volume (Å 95 95 24 94,5 94,5 20 94 94 (d) 93,5 93,5 16 93,59494,59595,596 93,59494,59595,596 U Np Am Cm α γ ε δ Pu GGA: Cohesion is overestimated, not enough correlation GGA(AFM) G. Robert, A. Pasturel, and B. Siberchicot et al Journal of Phys: Cond. Matter 15 8377 (2003), A. Kutepov and S. Kutepova J. Magn. Magn. Mater. 272, E329 (2004) GGA+OP P . S¨ oderlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P . S¨ oderlind al MRS Bull. 35, 883 (2010) LDA+GA N. Lanat` a, Y. Yao, C.-Z. Wang, K.-M. Ho, and G. Kotliar Phys. Rev. X 5, 011008 (2015) GGA+DMFT B. Amadon, Phys. Rev. B 94 , 115148 (2016) — Conference: What about U ? — 5/38
GGA-AFM improves volumes 32 32 Experiment GGA (AFM) GGA+OP (AFM) GGA (NM) 28 28 3 ) Volume (Å 24 24 20 20 (a) (b) (d) 16 16 Am Cm Am Cm U Np Am Cm α γ εδ α γ εδ α γ ε δ Pu Pu Pu GGA-AFM: good description of volumes but magnetism is wrong GGA(AFM) G. Robert, A. Pasturel, and B. Siberchicot et al Journal of Phys: Cond. Matter 15 8377 (2003), A. Kutepov and S. Kutepova J. Magn. Magn. Mater. 272, E329 (2004) GGA+OP P . S¨ oderlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P . S¨ oderlind al MRS Bull. 35, 883 (2010) LDA+GA N. Lanat` a, Y. Yao, C.-Z. Wang, K.-M. Ho, and G. Kotliar Phys. Rev. X 5, 011008 (2015) GGA+DMFT B. Amadon, Phys. Rev. B 94 , 115148 (2016) — Conference: What about U ? — 6/38
LDA+GA results 32 32 Experiment GGA (AFM) LDA+GA (NM) GGA+OP (AFM) GGA (NM) 28 28 3 ) Volume (Å 24 24 20 20 (a) (b) (c) (d) 16 16 Am Cm Am Cm Am Cm U Np Am Cm α γ εδ α γ εδ α γ εδ α γ ε δ Pu Pu Pu Pu Gutzwiller with U =4.5 eV, J H =0.36 eV: good volumes and magnetism GGA(AFM) G. Robert, A. Pasturel, and B. Siberchicot et al Journal of Phys: Cond. Matter 15 8377 (2003), A. Kutepov and S. Kutepova J. Magn. Magn. Mater. 272, E329 (2004) GGA+OP P . S¨ oderlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P . S¨ oderlind al MRS Bull. 35, 883 (2010) LDA+GA N. Lanat` a, Y. Yao, C.-Z. Wang, K.-M. Ho, and G. Kotliar Phys. Rev. X 5, 011008 (2015) GGA+DMFT B. Amadon, Phys. Rev. B 94 , 115148 (2016) — Conference: What about U ? — 7/38
GGA+DMFT on Pu Pioneering DFT+DMFT calculation with U =4 eV and J H =0 eV. (S. Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)) — Conference: What about U ? — 8/38
Is it possible to describe all actinides with a single framework ? Calculation of effective interactions U and J H with cRPA ? Does DFT+DMFT calculations with cRPA interactions give good structural properties ? What is the role of crystal structure ? Role of SOC and Hund’s exchange J H ? — Conference: What about U ? — 9/38
Self-consistent calculation of U in ABINIT cRPA calculation Compute cRPA dielectric matrix χ r 0 ( r , r ′ , ω ) = X ψ σ ∗ ν ′ k ′ ( r ) ψ σ ∗ ν ′ k ′ ( r ′ ) ψ σ ν k ( r ′ ) ν k ( r ) ψ σ k , k ′ ,ν,ν ′ ,σ f σ ν ′ k ′ − f σ × w ( k , k ′ , ν, ν ′ , σ ) ν k ν k + ω + i δ . ν ′ k ′ − ǫ σ ǫ σ "X # "X # DFT+ U calculation w ( k , k ′ , ν, ν ′ , σ ) = 1 − m k �| 2 m ′ k ′ �| 2 |� Ψ σ ν k | w σ |� Ψ σ ν ′ k ′ | w σ m m ′ Diagonalize H ε r ( ω ) = 1 − v χ r 0 ( ω ) . ⇒ ǫ σ k ,ν ⇒ | Ψ σ k ν � Compute PLO-Wannier Build Hamiltonian P R σ m ν ( k ) = � w R σ k m | Ψ k ν � H U , J H [ n ( r )] | w R σ X k ν � P R σ k m � = | Ψ σ m ν ( k ) ν ∈W Compute density X n ( r ) = Ψ σ k ν ( r ) f σ ν k Ψ σ k ν ( r ) ν, k ,σ Compute effective interaction matrix U σ,σ ′ m 1 w R σ ′ m 2 w R σ ′ m 1 , m 3 , m 2 , m 4 ( ω ) = � w R σ m 3 | ε − 1 ( ω ) v | w R σ m 4 � r Compute static U and J H 2 l + 1 2 l + 1 U = 1 1 U σ,σ ′ X X X m 1 , m 2 , m 1 , m 2 ( 0 ) 4 ( 2 l + 1 ) 2 σ,σ ′ m 1 = 1 m 2 = 1 U , J H 2 l + 1 2 l + 1 J H = 1 1 X X X U σ,σ ′ m 1 , m 2 , m 2 , m 1 ( 0 ) 4 ( 2 l + 1 )( 2 l ) σ,σ ′ m 1 = 1 m 2 = 1 ( m 2 � = m 1 ) F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein PRB 70, 195104 (2004) K. Karlsson, F. Aryasetiawan, and O. Jepsen Phys. Rev. B 81, 245113 (2010) B. Amadon, T. Applencourt, F. Bruneval, Physical Review B 89 , 125110 (2014) — Conference: What about U ? — 10/38
What about U ? U is small for plutonium and americium U Np Pu Am Cm 0.8 1.0 0.95 1.5 3.4 U U diag 1.5 1.7 1.7 2.3 4.3 0.4 0.4 0.45 0.4 0.55 J H U is weaker than expected in Pu and Am — Conference: What about U ? — 11/38
DFT+DMFT in ABINIT DFT+DMFT Loop DFT DFT Compute PLO-Wannier Diagonalize H P R m ν ( k ) = � w R k m | Ψ k ν � ⇒ ǫ k ν | w R X P R k m � = m ν ( k ) | Ψ k ν � ⇒ | Ψ k ν � ν ∈W Build Hamiltonian DMFT Loop DMFT Loop H [ n ( r )] Compute lattice Green’s function ∆Σ R , imp ( i ω n ) = Σ R , imp ( i ω n ) − Σ R dc . mm ′ mm ′ Compute density ν m ( k ) ∆Σ R , imp ∆Σ bl νν ′ k ( i ω n ) = X X P R ∗ ( i ω n ) P R m ′ ν ′ ( k ) mm ′ X Ψ k ν ( r ) G τ = 0 − n ( r ) = νν ′ k Ψ k ν ′ ( r ) R mm ′ i − 1 νν ′ k h G bl νν ′ k ( i ω n ) = ( i ω n + µ − ǫ ν k ) δ νν ′ − k ( i ω n ) bl νν ′ Compute Fermi level µ Compute local quantities ν ′ m ′ ( k ) − Σ R dc − µ ǫ R , imp X P R m ν ( k ) ǫ ν k P R ∗ = mm ′ k ,νν ′ Rotate quantities in the basis where ǫ R , imp is diagonal mm ′ G R , imp X P R m ν ( k ) G bl νν ′ k ( i ω n ) P R ∗ ( i ω n ) = ν ′ m ′ ( k ) mm ′ k ,νν ′ ( i ω n )] − 1 = [ G R , imp ( i ω n )] − 1 + Σ R , imp [ G 0 R , imp ( i ω n ) mm ′ mm ′ mm ′ F R mm ′ ( i ω n ) = i ω n − ǫ R m − [ G 0 R , imp ( i ω n )] − 1 m , m ′ m , m ′ Compute Self-energy Impurity Solver (CTQMC) G R , imp mm ′ ( i ω n ) m , U R ⇒ G R F R mm ′ ( τ ) , ǫ R mm ′ ( τ ) Σ R , imp mm ′ ( i ω n ) B. Amadon Journal of Phys.: Cond. Matter 24 , 075604 (2012). J. Bieder, B. Amadon, Phys. Rev. B 89 , 195132 (2014) — Conference: What about U ? — 12/38
U and J H are used to compute structural properties in DFT+DMFT. 32 32 Experiment GGA (AFM) LDA+GA (NM) GGA+OP (AFM) GGA+DMFT (NM) GGA (NM) 28 28 3 ) Volume (Å 24 24 20 20 (a) (b) (c) (d) 16 16 Am Cm Am Cm Am Cm U Np Am Cm α γ εδ α γ εδ α γ εδ α γ ε δ Pu Pu Pu Pu A good description of structural and magnetic properties GGA(AFM) G. Robert, A. Pasturel, and B. Siberchicot et al Journal of Phys: Cond. Matter 15 8377 (2003), A. Kutepov and S. Kutepova J. Magn. Magn. Mater. 272, E329 (2004) GGA+OP P . S¨ oderlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P . S¨ oderlind al MRS Bull. 35, 883 (2010) LDA+GA N. Lanat` a, Y. Yao, C.-Z. Wang, K.-M. Ho, and G. Kotliar Phys. Rev. X 5, 011008 (2015) GGA+DMFT B. Amadon, Phys. Rev. B 94 , 115148 (2016) — Conference: What about U ? — 13/38
Comparison with previous works 3 ) V ( ˚ U , J H (eV) B 0 (GPa) A δ -Plutonium GGA 0.00, 0.00 19.8 GGA+DMFT Savrasov et al (2001) 4.00, 0.00 26.5 GGA+DMFT/CTQMC (this work) 4.00, 0.00 25.8 28 — Conference: What about U ? — 14/38
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