[PPT] - Ab initio description of actinides: Hunds exchange, spin-orbit PowerPoint Presentation

SLIDE 1

www.cea.fr

Ab initio description of actinides: Hund’s exchange, spin-orbit coupling and crystal structure

Bernard Amadon

CEA, DAM, DIF , Arpajon, France

Conference: What about U ?

SLIDE 2

Actinides in the periodic table

4d element: filling of the 4d band (Bonding states and antibonding): 4d electrons are delocalized. Lanthanides: 4f electrons are localized , negligible

verlap

between 4f

rbitals .

Actinide: intermediate case

f

localization.

At atmospheric pressure: [Mac Mahan, et al J. Comp.-Aid. Mater. Des. 5, 131 (1998)] — Conference: What about U ? — 2/38

SLIDE 3

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 Instability.

Los Alamos Science

1.000 1.061 1.125 1.191 1.260 2 4 Pressure (kbar) 6 8 1.00 1.02 1.04 1.06 1.08 200 400 T e m p e r a t u r e ( ° C ) 600 2 4 6 8 1 Pressure (kbar) 2 4 6 T e m p e r a t u r e ( ° C ) V

l

u m e Length

α β γ δ δ′ ε ζ Figure 16. Plutonium Instability with Temperature and Pressure

Plutonium is notoriously unstable under almost any external disturbance. Over a span

f 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.)

From Kevin T. Moore and Gerrit van der Laan Rev. Mod. Phys. 81, 235 (2009) — Conference: What about U ? — 3/38

SLIDE 4

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

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Non magnetic GGA underestimates volume for late actinides

93,59494,59595,596 93,5 94 94,5 95 95,5 96 Volume (Å

3)

93,59494,59595,596 93,5 94 94,5 95 95,5 96 U Np

α

Cm Am

γ ε δ

16 20 24 28 32

Experiment GGA (NM)

Pu (d)

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¨

derlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P

. S¨

derlind 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

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GGA-AFM improves volumes

α

Cm Am

γ εδ GGA+OP (AFM) α

Cm Am

γ εδ

16 20 24 28 32 Volume (Å

3) GGA (AFM)

U Np

α

Cm Am

γ ε δ

16 20 24 28 32

Experiment GGA (NM)

Pu Pu Pu (a) (b) (d)

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¨

derlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P

. S¨

derlind 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

SLIDE 7

LDA+GA results

α

Cm Am

γ εδ

LDA+GA (NM)

α

Cm Am

γ εδ GGA+OP (AFM) α

Cm Am

γ εδ

16 20 24 28 32 Volume (Å

3) GGA (AFM)

U Np

α

Cm Am

γ ε δ

16 20 24 28 32

Experiment GGA (NM)

Pu Pu Pu Pu (a) (b) (c) (d)

Gutzwiller with U=4.5 eV, JH=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¨

derlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P

. S¨

derlind 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

SLIDE 8

GGA+DMFT on Pu Pioneering DFT+DMFT calculation with U=4 eV and JH=0 eV.

(S. Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)) — Conference: What about U ? — 8/38

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Is it possible to describe all actinides with a single framework ? Calculation of effective interactions U and JH 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 JH ?

— Conference: What about U ? — 9/38

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Self-consistent calculation of U in ABINIT

DFT+U calculation cRPA calculation

Diagonalize H ⇒ ǫσ

k,ν

⇒ |Ψσ

kν

Build Hamiltonian HU,JH[n(r)] Compute density

n(r) = X

ν,k,σ

Ψσ

kν(r)f σ νkΨσ kν(r)

Compute cRPA dielectric matrix

χr

0(r, r′, ω) =

X

k,k′,ν,ν′,σ

ψσ∗

νk (r)ψσ ν′k′(r)ψσ∗ ν′k′(r′)ψσ νk(r′)

× w(k, k′, ν, ν′, σ) f σ

ν′k′ − f σ νk

ǫσ

ν′k′ − ǫσ νk + ω + iδ .

w(k, k′, ν, ν′, σ) = 1 − "X

m

|Ψσ

νk|wσ mk|2

# "X

m′

|Ψσ

ν′k′|wσ m′k′|2

# εr(ω) = 1 − vχr

0(ω).

Compute PLO-Wannier

PRσ

mν(k) = wRσ km |Ψkν

|wRσ

km =

X

ν∈W

|Ψσ

kνPRσ mν(k)

Compute effective interaction matrix

Uσ,σ′

m1,m3,m2,m4(ω) = wRσ m1 wRσ′ m3 |ε−1 r

(ω)v|wRσ

m2 wRσ′ m4

Compute static U and JH

U = 1 4 X

σ,σ′

1 (2l + 1)2

2l+1

X

m1=1 2l+1

X

m2=1

Uσ,σ′

m1,m2,m1,m2(0)

JH = 1 4 X

σ,σ′

1 (2l + 1)(2l)

2l+1

X

m1=1 2l+1

X

m2=1(m2=m1)

Uσ,σ′

m1,m2,m2,m1(0)

U, JH

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

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What about U ? U is small for plutonium and americium

U Np Pu Am Cm U 0.8 1.0 0.95 1.5 3.4 Udiag 1.5 1.7 1.7 2.3 4.3 JH 0.4 0.4 0.45 0.4 0.55 U is weaker than expected in Pu and Am

— Conference: What about U ? — 11/38

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DFT+DMFT in ABINIT

DFT DMFT Loop DFT+DMFT Loop DFT DMFT Loop

Diagonalize H ⇒ ǫkν ⇒ |Ψkν Build Hamiltonian H[n(r)] Compute density

n(r) = X

νν′k

Ψkν(r)Gτ=0−

νν′k Ψkν′(r)

Compute PLO-Wannier

PR

mν(k) = wR km|Ψkν

|wR

km =

X

ν∈W

PR

mν(k)|Ψkν

Compute lattice Green’s function

∆ΣR,imp

mm′

(iωn) = ΣR,imp

mm′

(iωn) − ΣRdc . ∆Σbl

νν′k(iωn) =

X

R

X

mm′

PR∗

νm(k) ∆ΣR,imp mm′

(iωn) PR

m′ν′(k)

Gbl

νν′k(iωn) =

h (iωn + µ − ǫνk)δνν′ −

bl k (iωn)

i−1

νν′

Compute Fermi level µ

Compute local quantities

ǫR,imp

mm′

= X

k,νν′

PR

mν(k) ǫνk PR∗ ν′m′(k) − ΣRdc − µ

Rotate quantities in the basis where ǫR,imp

mm′

is diagonal GR,imp

mm′

(iωn) = X

k,νν′

PR

mν(k) Gbl νν′k(iωn) PR∗ ν′m′(k)

[G0 R,imp

mm′

(iωn)]−1 = [GR,imp

mm′

(iωn)]−1 + ΣR,imp

mm′

(iωn) F R

mm′(iωn) = iωn − ǫR m − [G0 R,imp m,m′

(iωn)]−1

m,m′

Impurity Solver (CTQMC) F R

mm′(τ) , ǫR m , UR ⇒ GR mm′(τ)

Compute Self-energy GR,imp

mm′ (iωn)

ΣR,imp

mm′ (iωn)

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

SLIDE 13

U and JH are used to compute structural properties in DFT+DMFT.

α

Cm Am

γ εδ

LDA+GA (NM)

α

Cm Am

γ εδ GGA+OP (AFM) α

Cm Am

γ εδ

16 20 24 28 32 Volume (Å

3) GGA (AFM)

U Np

α

Cm Am

γ ε δ

16 20 24 28 32

Experiment GGA+DMFT (NM) GGA (NM)

Pu Pu Pu Pu (a) (b) (c) (d)

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¨

derlind and B. Sadigh Phys. Rev. Lett. 92, 185702 (2004), P

. S¨

derlind al MRS Bull. 35, 883 (2010)

Role of interaction on phases of Pu with different structures.

Phase Number of neighbors dmin

Pu−Pu

α (pseudo-α) 4 2.51 δ 12 2.96

δ phase has more neighbors, but they are farther apart. α phase has less neighbors, but they are closer. Chemical bonding is larger in α phase. The α phase is less sensitive to interactions.

Pseudo-α phase: J. Bouchet, R. C. Albers, M. D. Jones, and G. Jomard Phys. Rev. Lett. 92, 095503 (2004). — Conference: What about U ? — 19/38

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Role of interaction on phases of Pu with different structures.

V E α GGA δ GGA α GGA+DMFT δ GGA+DMFT Chemical bonding stronger in the α phase, less sensitive to interactions. Chemical bonding weaker in the δ phase, more sensitive to interactions.

— Conference: What about U ? — 20/38

SLIDE 21

Role of interaction on phases of Pu with different structures.

V E α GGA δ GGA α GGA+DMFT δ GGA+DMFT α γ ε δ

16 16 20 20 24 24 28 28 32 32 Volume per atom (A

°3)

Experiment GGA+DMFT (NM) GGA (NM)

Plutonium

Chemical bonding stronger in the α phase, less sensitive to interactions. Chemical bonding weaker in the δ phase, more sensitive to interactions.

— Conference: What about U ? — 21/38

SLIDE 22

Role of interaction on phases of Pu with different structures.

V E α GGA

(a) (a)

δ GGA

(b) (b)

α GGA+DMFT δ GGA+DMFT (a) Chemical bonding stronger in α phase, less sensitive to interactions. (b) Electrons are more delocalized: interaction larger in the α phase.

— Conference: What about U ? — 22/38

SLIDE 23

Role of interaction on phases of Pu with different structures.

V E α GGA

(a) (a)

δ GGA

(b) (b)

α GGA+DMFT δ GGA+DMFT α γ ε δ

100 100 200 200 300 300 400 400 500 500 600 600 700 700 (meV)

GGA GGA+DMFT

Exp. analysis

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Electronic origin of the transition ? Role of SOC and JH ?

— Conference: What about U ? — 30/38

SLIDE 31

Role of spin-orbit coupling

α-Pu δ-Pu Cm Am 16 20 24 28 Volume (Å

3)

Experiment DFT+DMFT SOC and JH=0.45

SOC reduces the degeneracy of the f orbitals from 14 to 6 in the atomic limit: increase correlation effects. JH increases the filling of the J=5/2 shell: the effective degeneracy is closer to 6 than without JH.

— Conference: What about U ? — 31/38

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Spin-orbit coupling is mandatory !

α-Pu δ-Pu Cm Am 16 20 24 28 Volume (Å

3)

Experiment DFT+DMFT SOC and JH=0.45 DFT+DMFT no SOC, JH=0.45

SOC reduces the degeneracy of the f orbitals from 14 to 6 in the atomic limit: increase correlation effects. JH increases the filling of the J=5/2 shell: the effective degeneracy is closer to 6 than without JH.

— Conference: What about U ? — 32/38

SLIDE 33

Effect of SOC on f levels.

f levels

SOC

J = 7

2

J = 5

2

As in multibands Hubbard models, degeneracy is reduced, thus correlations are enhanced.

— Conference: What about U ? — 33/38

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Hund’s Exchange is important

α-Pu δ-Pu Cm Am 16 20 24 28 Volume (Å

3)

Experiment DFT+DMFT SOC and JH=0.45

SOC reduces the degeneracy of the f orbitals from 14 to 6 in the atomic limit: increase correlation effects. JH increases the filling of the J=5/2 shell: the effective degeneracy is closer to 6 than without JH.

— Conference: What about U ? — 34/38

SLIDE 35

Hund’s exchange is important

α-Pu δ-Pu Cm Am 16 20 24 28 Volume (Å

3)

Experiment DFT+DMFT SOC and JH=0.45 DFT+DMFT SOC and JH=0

JH increases the filling of the J=5/2 shell: the effective degeneracy is closer to 6 than without JH. SOC reduces the degeneracy of the f orbitals from 14 to 6 in the atomic

limit. to 6 than without JH.

— Conference: What about U ? — 35/38

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Role of JH in δ Plutonium

J = 7

2

J = 5

2

JH = 0 n 7

2 ≃ 0.5

n 5

2 ≃ 4.8

JH JH = 0.45 eV n 7

2 ≃ 0.2

n 5

2 ≃ 5.0

Hund’s coupling JH increases the polarization of 5

2 orbitals.

why ?

— Conference: What about U ? — 36/38

SLIDE 37

JH enhances the polarization of J=5/2 orbitals.

J = 7

2

J = 5

2 U 5

2 5 2

E = U 5

2 5 2

U 5

2 7 2

E = U 5

2 7 2 + λSOC

U 5

2 7 2 > U 5 2 5 2 ⇒ The weight of Slater determinants involving 7

2 states is

reduced not only by SOC but also by Hund’s interaction.

For interactions, see also J.-P . Julien, J.-X. Zhu, and R. C. Albers, Phys. Rev. B 77, 195123 (2008) — Conference: What about U ? — 37/38

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Conclusion

U and JH are computed: U is small The same theoretical description for low volume actinides and high volume actinides. Key role of SOC and Hund’s exchange.

B. Amadon, Phys. Rev. B 94, 115148 (2016)

Thanks to F. Bottin, J. Bouchet, C. Denoual, B. Dorado, F. Jollet, and G. Robert for useful discussions — Conference: What about U ? — 38/38