unlocking the mysteries of plutonium via hpc

76
Unlocking the mysteries of Plutonium via HPC C.A. Marianetti LLNL-PRES-406717 HPC Conference - 10/22/2008

Upload: others

Post on 06-Feb-2022

4 views

Category:

Documents


0 download

TRANSCRIPT

Unlocking the mysteries ofPlutonium via HPC

C.A. Marianetti LLNL-PRES-406717 HPC Conference - 10/22/2008

Collaborators and Funding

• Kristjan Haule - Rutgers University

• Gabriel Kotliar - Rutgers University

• Mike Flusss - LLNL

• Funded by LLNL

• Atlas supercomputer

Overview

• Introduction to strong correlations

• Formalism

→ Density functional theory (DFT)

→ Dynamical mean-field theory (DMFT)

→ DFT+DMFT

• Background of Plutonium

• Application to Plutonium

Electronic Many-body Hamiltonian

H =∑

σ

drψ+σ (r)[−▽2 + Vext(r) − µ]ψσ(r)

+ 12

σσ′

drdr′ψ+σ (r)ψ+

σ′(r′) 1r−r

′ψσ′(r′)ψσ(r).

ψ+σ′(r)ψσ(r′) + ψσ(r′)ψ+

σ′(r) = δσσ′δ(r − r′)

ψσ′(r)ψσ(r′) + ψσ(r′)ψσ′(r) = 0

Local, atomic-like basis setR → Lattice site α, β, γ, δ→ orbital type (ie.s,p,d,f )

ψ(x) =∑

αR cαR(τ)χαR(r)

H =∑

αβ

RR′ h(0)αRβR′

(

c+αRcβR′ + h.c.)

+

12

αβγδ

RR′R′′R′′′ V RR′R′′R′′′

αβγδ c+αRc+βR′cδR′′′cγR′′

where...

h(0)αRβR′ = 〈χαR| − ∇2 + Vext|χβR′〉

V RR′R′′R′′′

αβγδ = 〈χαR(r)χβR′(r′)| 1r−r′

|χγR′′(r′)χδR′′′(r)〉

Start with something simple...

• Ones orbital per lattice site...

• Only hopping between nearest neighbors...

• Only keep Coulomb repulsion ifR = R′ = R′′ = R′′′

→ Only for electrons on the same site...

• Simple model which retains relevant physics

• The Hubbard model

The Hubbard model

H =∑

ijσ

tijc+iσcjσ +

i

Uni↑ni↓

Hopping Repulsion

tij = 〈χs| − ∇2 + Vext|χs〉

U = 〈χs(r)χs(r′)| 1r−r′

|χs(r′)χs(r)〉

• niσ = c+iσcjσ

• c+iσcjσ + cjσc+iσ = δij ciσcjσ = −cjσciσ

• Hopping terms yields delocalized electrons

• Repulsion term yields localized electrons

Consider a sqaure lattice...perhaps a hydrogen on each vertex...

One electron per site...Large U/t prevents hopping→ Mott insulator

down up both

One electron per site...Large U/t prevents hopping→ Mott insulator

down up both

One electron per site...Large U/t prevents hopping→ Mott insulator

down up both

One electron per site...Large U/t prevents hopping→ Mott insulator

down up both

One electron per site...Large U/t prevents hopping→ Mott insulator

down up both

Hole in system...Hole can diffuse... even for U/t→∞

down up both

Hole in system...Hole can diffuse... even for U/t→∞

down up both

Hole in system...Hole can diffuse... even for U/t→∞

down up both

Hole in system...Hole can diffuse... even for U/t→∞

down up both

Hole in system...Hole can diffuse... even for U/t→∞

down up both

Dilute system...Delocalized... even for large U/t...

down up both

Dilute system...Delocalized... even for large U/t...

down up both

Dilute system...Delocalized... even for large U/t...

down up both

Strongly correlated electron systems

• Close to a Mott transition

• Mott insulator doped with holes

• High temperature superconductors

• Actinides

• Collosal mangeto-resistance materials

• transition-metal oxides

• many secrets locked in strongly correlated systems

Effects of electronic correlationsMagnetic Susceptibility

χ = ∂2E∂H2 = ∂M

∂H

•Metallic state - Fermi Liquid (FL)

→ interactions make electrons become HEAVY, butremain coherent

→ χ ∝ m∗ → temperature independent

• Localized state

→ interactions/temperature destroy FL!

→ χ ∝ 1T→ Curie

0 2 4 6 8 10Temperature

0

0.2

0.4

0.6

0.8

1

χ

Many-body Hamiltonian

H =∑

σ

drψ+σ (r)[−▽2 + Vext(r) − µ]ψσ(r)

+ 12

σσ′

drdr′ψ+σ (r)ψ+

σ′(r′)vC(r − r′)ψσ′(r′)ψσ(r).

Electronic structure calculations with DMFT

Reviews of Modern Physics, 78, 865 (2006)

http://dmft.rutgers.edu

General Methods

• Density functional theory

⇒ Realistic systems

⇒ Ground state properties

• Dynamical mean-field theory

⇒ Model Hamiltonians

⇒ Captures physics of Mott transition

⇒ Ground/excited state properties

• Merge approaches→ DFT+DMFT

Density Functional Theory

• E[ρ] is minimal whenρ is ground state density

• Kohn-ShamH0 + Vks(r)

• Vks(r) is what must be added toH0 to generate correctdensity

• Many-body problem partially mapped to one-bodyproblem butVks(r) not known!

• ApproximateVks with LDA

• Vks = ǫxc + ∂ǫxc

∂ρρ

Green’s (Function) Functional Theory

• Γ[G] is stationary whenG is the interacting Greensfunction

• DysonH0 + Σ(k, ω)

• Σ(k, ω) is what must be added toH0 to generatecorrect Greens function

• Many-body problem partially mapped to timedependent one-body problem butΣ(k, ω) not known!

• ObtainΣ(k, ω) using DMFT

What is DMFT?

• DMFT maps lattice→ impurity

Hlat =∑

ijσ

tijc+iσcjσ +

i

Uc+i↑ci↑c+i↓ci↓

Himp =∑

σ

ǫimpc+σ cσ + Uc+↑ c↑c

+↓ c↓ +

Vk(a+kσcσ + c+σ akσ) +

αka+k ak

• αk, Vk given by DMFT self-consistency

• Σlat(k, ω) ≈ Σlocal(ω)

DMFT algorithm

• guessαk, Vk → G0imp(ω)

• G0imp(ω) → impurity solver→ Gimp(ω),Σimp(ω)

• Σlat(k, ω) = Σimp(ω)

• Performself-consistency condition

Gimp′ =∑

k

Glat(k, ω) =∑

k

1ω−Hk+µ−Σimp(ω)

• G0imp(ω) = (Σimp(ω) +G−1

imp(ω))−1

→ αk, Vk

• iterate

Solving impurity problem3 main issues...

• ”Exact” but slow

• Fast but not always reliable

• real axis vs. imaginary axis

Solving impurity problem

• Hartree-Fock (DFT+DMFT→ DFT+U )

• FLEX - perturbation theory around band limit

• NCA - perturbation theory around atomic limit

• Exact Diagonalization

• Slave Boson mean-field

• Hirsch-Fye QMC

• Continuous time QMC

• NRG, DMRG, etc...

CTQMC

Merging DFT and DMFT

• Define local basis set

• Identify correlated subspace of Hamiltonian

⇒ Apply correlations∑

αβ

Uαβnαnβ

• Chooseρ andGloc as variables⇒ Γ[ρ,Gloc]

• H0 + VKS +∑

αβ

|α〉〈β|Σαβ(ω)

• DetermineU and ”double-counting” correction

• ObtainΣ using DMFT

• Perform both DMFT and DFT self-consistency

Applications of DFT+DMFT

• f -electron systems

• Transition-metal oxides

• Transition-metals

• Surfaces / Interfaces

• Multilayers

• Disorder

• Electron-phonon coupling

• etc, etc...

What can DFT+DMFT predict?

• Spectra

• Fermi Surface

• Density

• Total Energy

• Entropy

• Magnetic Susceptibility

• Heat capacity

• Transport

• etc, etc...

Application: Plutonium

Phase diagram of Pu

0 200 400 600 800 1000Temperature (K)

0

5

10

15

20

25

30

% V

olum

e E

xpan

sion

α

βγ

δ δ’ε L

Phase diagram of Pu

200 400 600 800 1000 1200 1400 1600 1800Temperature (K)

0

5

10

15

20

25

30

% V

olum

e E

xpan

sion

Pu

Fe

Compare Pu to other metals

200 400 600 800 1000 1200 1400 1600 1800Temperature (K)

0

5

10

15

20

25

30

% V

olum

e E

xpan

sion

Pu

Fe

Mn

α-Pu

β-Pu

γ-Pu

δ-Pu

δ’-Pu

ǫ-Pu

Volume vs. atomic number

Experimental Magnetic Susceptibility

100 200 300 400 500 600 700Temperature (Kelvin)

0

2

4

6

8

χ (1

03 em

u/m

ol)

UBe13

(m*/m~300)

Pu

Results forδ PuPRL 101, 056403 (2008)

24 26 28 30 32

Volume (A3)

0.1

0.2

0.3

0.4

Z

J=0.5eVJ=0.0

24 25 26 27 28 29 30 31 32 33

Volume (A3)

20

25

30

35

40

45

50

55

γ (m

J/m

ol-K

2 )

200 400 600 800 1000 1200T (Kelvin)

3

4

5

6

7

8

9

10

χ loca

l

V=24.8 A3

V=28.8 A3

V=32.8 A3

0

2

4

6

8

10

eV-1

S=5/2S=7/2

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

2

4

6

eV-1

S=5/2 J=0S=7/2 J=0S=5/2 J=0.5eVS=7/2 J=0.5eV

DFT

DMFT

• Performed approximation-free DMFT at ambient T

→ Including full exchange interaction

• Calculated various properties

→ Quasiparticle weight → Magnetic susceptibility

→ Heat capacity → Spectra

Photoemission spectra

0

2

4

6

8

10

eV-1

S=5/2S=7/2

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

2

4

6

eV-1

S=5/2 J=0S=7/2 J=0

DFT

DMFT

• Density of electronic statesversus energy.

• DMFT renormalizes DFTspectrum

→ Transfer of weight from Fermienergy to Hubbard bands

• Exchange causes furtherrenormalization of spectrum.

Photoemission spectra

0

2

4

6

8

10

eV-1

S=5/2S=7/2

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

2

4

6

eV-1

S=5/2 J=0.5eVS=7/2 J=0.5eV

DFT

DMFT

• Density of electronic statesversus energy.

• DMFT renormalizes DFTspectrum

→ Transfer of weight from Fermienergy to Hubbard bands

• Exchange causes furtherrenormalization of spectrum.

Photoemission spectra

0

2

4

6

8

10

eV-1

S=5/2S=7/2

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

2

4

6

eV-1

S=5/2 J=0S=7/2 J=0S=5/2 J=0.5eVS=7/2 J=0.5eV

DFT

DMFT

• Density of electronic statesversus energy.

• DMFT renormalizes DFTspectrum

→ Transfer of weight from Fermienergy to Hubbard bands

• Exchange causes furtherrenormalization of spectrum.

Exchange has notable effect on spectra

Photoemission

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

1

2

3

4

5

6

7

8eV

-1

S=5/2 J=0S=7/2 J=0S=5/2 J=0.5 eVS=7/2 J=0.5 eV

Photoemission Experiment

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2eV

0

4

8

12

16

20

Gouder et alTobin et al

Photoemission Comparison

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2eV

0

1

2

3

4

5

6

Gouder et alTotalspdf

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600KT=500K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400KT=352K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400KT=352KT=300K

Photoemission Temperature Dependence

-0.2 -0.1 0 0.1 0.2eV

0

0.2

0.4

0.6

0.8

1eV

-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400KT=352KT=300KT=240K

Quasiparticle WeightZ = mm∗

24 26 28 30 32

Volume (A3)

0.1

0.2

0.3

0.4

Z

J=0.5eVJ=0.0

• Z is inverse of effective mass

→ Z=1⇒ no correlations

→ Z=0⇒ electrons localize

• Pu is closer to atomic limit

• Electrons become heavier asvolume increases

• Including exchange substantiallyincreases correlations.

δ Pu is strongly correlated

Linear Coefficient of Heat Capacity

24 25 26 27 28 29 30 31 32 33

Volume (A3)

20

25

30

35

40

45

50

55

γ (m

J/m

ol-K

2 )

• Low temperature heat capacitysensitive to correlations.

• Strongly depends on volume.

• Experiments find35 − 65 mJmolK2

→ Huge expt. variation

• Cause of discrepancy:

→ Electron-phonon coupling

→ Density self-consistency

→ number of f electrons

→ Inaccurate experiments?γ =

2πk2

B

3

αρα(0)Zα

Magnetic Susceptibilityχ

200 400 600 800 1000 1200T (Kelvin)

3

4

5

6

7

8

9

10

χ loca

l

V=24.8 A3

V=28.8 A3

V=32.8 A3

• First calculation ofχ in Pu

• Pauli behavior⇒ itinerantelectrons

• Curie behavior⇒ localizedelectrons

• Predict Pauli behavior forVeq

→ Agrees with experiments

→ Explains lack of magnetism

• Expanded lattice agrees withPuH2

Quantum decoherence inδ Pu

Experimental Magnetic Susceptibility

0 50 100 150 200 250 300Temperature (Kelvin)

0

0.001

0.002

0.003

0.004

0.005em

u/m

ole

δ Pu

PuH2

The effect of Structure on Correlations

• Eplore differences betweenδ andǫ

→ FCC vs. BCC

• Use the same volume to isolate effect of structure

Photoemissionǫ-Pu

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

2

4

6

8eV

-1

S=5/2S=7/2

Photoemissionǫ-Pu vs.δ-Pu

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV

0

1

2

3

4

5

6

7

8eV

-1

S=5/2 εS=7/2 εS=5/2 δS=7/2 δ

Quasiparticle WeightZ = mm∗

24 26 28 30 32

Volume (A3)

0.1

0.2

0.3

0.4

Z

J=0.5eVJ=0.0J=0.5eV ε

• Z is inverse of effective mass

→ Z=1⇒ no correlations

→ Z=0⇒ electrons localize

• Pu is closer to atomic limit

• Electrons become heavier asvolume increases

• Including exchange substantiallyincreases correlations.

δ Pu is strongly correlated

Magnetic Susceptibilityχ

200 400 600 800 1000 1200T (Kelvin)

3

4

5

6

7

8

9

10

χ loca

l

V=24.8 A3

V=28.8 A3

V=32.8 A3

ε-Pu V=24.8 A3

• First calculation ofχ in Pu

• Pauli behavior⇒ itinerantelectrons

• Curie behavior⇒ localizedelectrons

• Predict Pauli behavior forVeq

→ Agrees with experiments

→ Explains lack of magnetism

• Expanded lattice agrees withPuH2

Quantum decoherence inδ Pu

Conclusions

• CTQMC allows for an accurate treatment of Pu

→ ambient temperatures

→ rotationally invariant exchange interaction

• Pu displays Pauli behavior in magnetic susceptibility

• Pu undergoes electronic decoherence as the volume isexpanded

• Correlations similar for FCC and BCC