First-Principles Study of Fe Spin Crossover in the Lower Mantle

Dane Morgan, Amelia BengtsonMaterials Science and EngineeringUniversity of Wisconsin – Madison

Second VLab WorkshopUniversity of Minnesota

August 5-10, 2007

Computational Materials GroupUniversity of Wisconsin - Madison

Faculty− Dane Morgan− Izabela Szlufarska

Graduate Students− Amelia (Amy) Bengtson− Edward (Ted) Holby− Trenton Kirchdoefer− Yueh-Lin Lee− Yun Liu− Yifei Mo− Julie Tucker− Marcin Wojdyr− Benjamin (Ben) Swoboda

Undergraduates− Paul Kamenski

http://matmodel.engr.wisc.edu/

Please stop by Amy’s poster!!

Outline

Fe and Spin Crossover in the Lower Mantle

First-Principles Modeling: Opportunities and Challenges

First-Principles study of Fe Spin Crossover

− Composition effects

− Volumes effects

− Structural effect: Ferropericlase vs. perovskite

Fe and Spin Crossover in the Lower Mantle

The Lower Mantle

Largest continuous region of Earth (~50% mass/volume)

Depth ≈ 660 – 2690 km T ≈ 2000-4000 K P ≈ 25-135 GPa Made of

− (Mg,Fe,Al)(Si,Al)O3 perovskite (62%)

− (Mg,Fe)O ferropericlase (rocksalt) (33%)− (Mg,Fe)(Si,?)O3 post-perovskite (>125

Gpa) Murakami, et al., Science ‘04

− cFe/(cMg+cFe) ~ 0.2

− CaSiO3 (5%)

− Impurities (~0%) Jackson and Ridgden '98 Duffy, Nature ‘04

Octahedral Fe2+ Spin State

Intermediate spinM = 2B

Low spinM = 0B

eg

t2g

High spinM = 4B

Majority

Minority

Exf

EHund

Spin State of Fe in the Lower Mantle: Ferropericlase

X-ray emission spectra, Mg0.83Fe0.17O

P = 0 GPa high spin

P = 75 GPa low spin

Badro, et al., Science ‘03

Spin State of Fe in the Lower Mantle: Perovskite

X-ray emission spectra, perovskite

(Mg0.87Fe0.09)(Si0.94Al0.10)O3(Mg0.92Fe0.09)Si1.00O3

P = 2 GPa high spin

P = 100 GPa intermediate spin

Li, et al., PNAS ‘04

Spin State vs. Temperature: (Mg0.75,Fe0.25)O

Lin, et al., Science TBP

High vs. Low Spin - Does it Matter? YES!

Density: RHS = 0.78Å, RLS = 0.61Å (~25% change!) (Shannon, Acta Cryst. A ’76)

Composition: changes in spin could dramatically change Fe partitioning

Phase stability: spin transitions could couple to phase stability

Thermal transport: Optical absorption change change in radiative heat transfer properties

Thermoelasticity: Elastic constants could be very different – unknown at present

Kinetics, …

Fe spin in the Lower Mantle: Questions

How does spin state depend on− Pressure− Temperature− Composition− Local chemical order (Mg vs. Fe, Al neighbors)− Structure (rocksalt, iB8, perovskite, post-perovskite)− Fe valence (2+ vs. 3+)− Fe site occupancy (A, B site in perovskite)

How does the spin state impact− Fe partitioning− Lower mantle phase stability− Thermophysical properties (density, mechanical properties, heat

transport, etc.)

First-Principles Modeling: Opportunities and Challenges

Composition and Structure(e.g., Mg0.75Fe0.25O)

• Energies: Stability, Atomic Positions, …• Electronic Structure: Spin state, Bands, …• Additional modeling for T>0, optical properties, …

Quantum mechanics

(+ approximations)

First-Principles Calculations

First-Principles Approach

Broad technique: Density Functional Theory Exchange correlation: LDA, GGA, LDA+U, GGA+U

approaches Pseudopotentials: Ultrasoft pseudopotentials, Projector

Augmented Wave Method Relaxation: Full relaxation with symmetry perturbed

structures Numerics: meV/atom accuracy convergence of relative

energies with respect to kpoints and energy cutoff Disorder: Special Quasirandom Structures for

configurationally and magneticaly disordered cells (Wei, et al.,

PRB ‚90)

VASP code

Opportunities for First Principles and Spin Effects

How does spin state depend on− Pressure− Temperature− Fe composition− Structure (rocksalt, iB8, perovskite, post-

perovskite)− Fe valence (2+ vs. 3+)− Fe site occupancy (A, B site in perovskite)− Local chemical order (Mg vs. Fe, Al

neighbors) How does the spin state impact

− Fe partitioning− Lower mantle phase stability− Thermophysical properties (density,

mechanical properties, heat transport, etc.)

Can be obtained from first-principles or first-principles + modeling

Calculating Spin-Transitions

HSLS

V

E

VHS VLS

P

H=HHS–HLS

HS

LS

PT

First-Principles Prediction – (Mg,Fe)O

Lin, et al., Science TBPTsuchiya, et al., Phys. Rev. Lett. ‘06

CFe = 19%, Theory, 2006 CFe = 25%, Expt, 2007

First-Principles Fe-Spin Results

Spin state− HS state for iB8 in lower mantle (Persson, et al., Geo. Res. Lett. ‘06)− HS state for post-perovskite in lower mantle (Zang and Oganov, EPSL ’06,

Stackhouse, et al., Geo. Res. Lett. ‘06) − LS state for B-site Fe in perovskite in lower mantle (Cohen, et al., Science ‘97)

Crossover trends with composition, local order, valence, temperature− Increasing crossover pressure with increasing Fe content for (Mg,Fe)O (Persson,

et al., Geo. Res. Lett. ‘06)− Decreasing crossover pressure with increasing Fe content for (Mg,Fe)SiO3

(Bengtson, et al., Submitted)− Increasing crossover pressure for Fe3+ vs. Fe2+ (Li, et al., Geo. Res. Lett. ’05)− Increasing crossover pressure with increasing temperature (Tsuchiya, et al.,

Phys. Rev. Lett. ‘06)− Decreasing crossover pressure from local Fe neighbors in perovskite

(Stackhouse, et al., Geo. Res. Lett. ‘06)− Decreasing of crossover pressure with local Al neighbors in perovskite (Li, et al.,

Geo. Res. Lett. ’05) Spin effects

− Changes in optical properties (Tsuchiya, et al., Phys. Rev. Lett. ‘06)− Changes in volume, elastic constants (Persson, et al., Geo. Res. Lett. ‘06)

(apologies to those I missed!)

Challenges for First-Principles and Spin Effects

Why so much spread in

calculation?

Challenges for First-Principles and Spin Effects

Accuracy of calculation parameters− Exchange-correlation type: LDA/GGA− Exchange-correlation parametrization: PW, PBE, …− Correlated electron corrections: LDA/GGA+U− Pseudopotentials: All electron, Ultrasoft, PAW, …

Correct materials system parameters− Composition: global and local chemical order− Valence− Site occupancy− Temperature− Structural relaxation− Magnetism

Spin Transition Calculations Sensitivity: Calculation Parameters - (Mg0.75Fe0.25)SiO3

PT

200 GPa

150 GPa

100 GPa

GGA

GGA-PW (Perdew, et al. PRB ’92)

GGA-PBE (Perdew, et al. PRL ’97)

Exchange-correlation effects

LDA

Sensitivity to calculation method - which is best?

Spin Transition Calculations Sensitivity: Materials Parameters - (Mg0.75Fe0.25)SiO3

PT

200 GPa

170 GPa

140 GPa

dFe-Fe = 4.98 Ǻ

dFe-Fe = 3.38 Ǻ

Fe2+

GGA-PBE (Perdew, et al. PRL ’97)

Fe local order Valence effectAl local order

Fe3+ + Al

Sensitivity to valence/configurations – need to compare like configurations

Spin Transition Calculations Sensitivity: Materials Parameters - FeSiO3

PT

900 GPa

240 GPa

77 GPa

Cubic symmetry(Cohen, et al. Science ’92)

MgSiO3 symmetry(Stackhouse, et al. EPSL ’07)

No symmetry(Bengtson, et al. Submitted)

Structural relaxations

Sensitivity to structural relaxations – need to compare identical structures

Scale of Different Sensitivities

Calculation parameters− Exchange correlation type (LDA/GGA) ~100 GPa

− Exchange correlation parametrization ~30 GPa

− Pseudopotential choice ~30 GPa

− Correlation corrections (LDA+U) ~50 GPa

Materials system parameters− Structural relaxation ~1000 GPa

− Compositions ~100 GPa

− Local chemical ordering ~30 GPa

− Valence (Fe2+ vs. Fe3+) ~30 GPa

− Magnetic ordering ~30 GPa

Sensitivities ≠ Errors!Need good choices!

Summary of First-Principles Challenges

Comparing calculations: Equivalent materials systems and calculation parameters

Comparing experiments: Equivalent materials systems and best calculation parameters

Still learning!

First-Principles study of Fe Spin Crossover

Our Questions

What is the composition dependence of the spin crossover?

What drives the crossover – electronic vs. volume changes?

What differences might exist between ferropericlase (rocksalt) and perovskite structures?

Ferropericlase (Rocksalt)

(Mg,Fe)O Rocksalt structure

Fe octahedrally coordinated

Mg-Fe pseudobinary alloy on metal FCC sublattice

Generally assumed to be single disordered phase under lower mantle conditions

Ferropericlase

Strong composition -spin crossover coupling

What drives the crossover?

What drives composition effect?

Persson, et al., GRL ‘06

Ferropericlase: What Drives the Crossover?

E does not go to zero!

PV term is the most important driver of the transition!

Both E, PV terms drive up crossover pressure with Fe content

Effect of chemical pressure?

P∆V

∆E

Spin crossover (T=0) when H = EHS-ELS + P(VHS-VLS) = 0

Understanding PT vs. CFe TrendChemical Pressure

0

40

80

120

160

200

0 0.2 0.4 0.6 0.8 1Fe Concentration

Pressure (GPa)

6.8

7

7.2

7.4

7.6

7.8

Volume (A

3 /atom)

PT

▲Volume (P=100GPa)HS

LS

Mg compresses Fe-HS HS less stable PT↓ Mg does not expand Fe-LS LS unaffected PT↔ Increasing Mg pushes PT↓

P=0: R(Fe-HS)>R(Mg)≈R(Fe-LS)

Perovskite

(Mg,Fe)(Si)O3 perovskite structure

Fe in pseudocubic environment

Mg-Fe pseudobinary alloy on metal cubic sublattice

Generally assumed to be single disordered phase under lower mantle conditions for low Fe content, unstable for high Fe content

Perovskite

Bengtson, et al., EPSL, submitted ‘07

Strong composition -spin crossover coupling, opposite ferropericlase!

What drives the crossover?

What drives composition effect?

Perovskite: What Drives the Crossover?

P∆V

∆E

PV still very important in transition

E terms drive down crossover pressure with Fe content

Changes in E due to structural relaxations (crossover pressure = ~900 GPa w/o relaxation!)

Crossover Pressure vs. Fe Composition Strong Structural Coupling

Transitions driven significantly by PV terms Opposite trends due to structural relaxation in perovskite

Ferropericlase

Perovskite

Conclusions

Wide range of spin crossover values possible with different calculation and system choices.

Spin crossover trends with composition are opposite in ferropericlase and perovskite.

Volume contraction (PV) makes a major contribution to the spin crossover energetics.

Ferropericlase

Perovskite

P∆V

∆E

Acknowledgements

Additional collaborators: Jie Li (UIUC)

Funding: Wisconsin Alumni Research Foundation (WARF)

End

Ferropericlase (Rocksalt)

(Mg,Fe)O Rocksalt structure Fe octahedrally coordinated Mg-Fe pseudobinary alloy on metal FCC

sublattice Phase stability: High T,P experiments

ambiguous:− Mg0.5Fe0.5O, Mg0.6Fe0.4O, Mg0.8Fe0.2O:

Phase separation (Dubrovinsky, et al., '00,'01,’05)

− Mg0.6Fe0.4O: No separation (Vissiliou and Ahrens,

Geophys. Res. Lett. ’82)

− Mg0.25Fe0.75O, Mg0.39Fe0.61O: No separation (Lin, et al., PNAS '03)

− Often assumed to be single disordered phase under lower mantle conditions for most compositions

A Multiscale Alloy Theory Approach

Multiscale Alloy Theory Approach

First-Principles Energetics

( ) 0, ,{ } T conf mag vib elecF P T c U PV F F F F== + + + + +

ThermodynamicModeling

Phase stability, Fe partitioning,Fe spin states, Densities, …

CALPHAD

Multiscale Alloy Theory Approach - What is Needed?

Identifying key interactions (T=0, P>0)

− Spin state vs. structure (rocksalt vs. perovskite)

− Spin state vs. Fe composition

− Fe – Mg interaction vs. spin state

− Fe spin state vs. valence (Fe2+ vs. Fe3+)

Thermodynamic models (T>0) Phase stability studies + integration with

experimental data

Fe – Mg Interaction vs. Spin State: Perovskite

Fe(low spin)-Mg alloy could be below miscibility gap in lower mantle Possible Fe solubility constraints, even for cFe/(cMg+cFe) ~ 0.1 Possibly strong clustering short-range-order

Tc(100GPa)≈900K Tc(100GPa)≈4500K

Low SpinHigh Spin

First-Principles study of Fe Spin Crossover T>0

First-Principles Model for Ferropericlase

Treat system as a ternary alloy – {c} = cMg, cFe-HS, cFe-LS

Consider only solid solution phases on B1 (NaCl) and iB8 (inverse-NiAs) (Fang, et al., Phys Rev. Lett. ’98)

Use first-principles based model to get F(P,T,{c}) and construct a phase diagram

MgO

FeO-HS FeO-LS

Free Energy Model

( )( )

( )( )1 21 3 1 3

First-Principles energies, SQS to simulate disorder

ln ln ln ln

ln 5

3-3 4 3 log ; 0.617

4

dis

conf Mg Mg Fe Fe Fe Fe HS Fe HS Fe LS Fe LS

mag Fe HS

vib D DB

ele

U

TS kT c c c c c c c c c

F c kT

h BF kT T T T

k M

F

π

− − − −

−

=

′ ′ ′ ′⎡ ⎤− = + + +⎣ ⎦= −

⎛ ⎞Ω⎛ ⎞= + = ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

( ) 2gln 3 (3 degenerate t states)c Fe HSc kT−= −

1. S. H. Wei, et al., Phys. Rev. B, '902. A. van de Walle and G. Ceder, Reviews of Modern Physics, '023. G. R. Burns, Minerological Applications of Crystal Field Theory, '93

First-principles

First-principles

Fitting The Free Energy

AnalyticFit and interpolate

MgO

FeO-HS FeO-LS

Set grid of fitting points in V, {c} space

Fit Udis(V) to Birch-Murnaghan equation of state

F to polynomial in {c} at a given P, T

i i ij i j ijk i j ki i j i j k

F Fc F c c F c c c≠ ≠ ≠

= + +∑ ∑ ∑

( ), ,{ } dis vib conf mag elecF P T c U PV F TS F F= + + − + +

Fitting The Free Energy

Fitting grid for B1 (NaCl)− Mixed spin data uncertain

so assume no Fe-HS – Fe-LS interaction

Fitting grid for iB8 (i-NiAs)− Ab initio almost no LS− Ab initio almost no Mg

solubility Easy to fit!

MgO

FeO-HS FeO-LS

MgO

FeO-HS FeO-LS

Phase DiagramC

Fe-H

S /CF

e

700

4000

Dep

th (

km)

CFe0-MgO 1-FeO

2-phase

iB8 HSB1 mixed spin

HS

LS

1800

2900

0.25 0.75

P≈140GPa

P≈30GPa

Development of CALPHAD Approach

Established collaboration with CompuTherm LLC− Makers of Pandat phase diagram software− Developing module to integrate our free energy functions into their

phase diagram solvers− Will allow far more complex phase diagram calculations, automated

free energy model optimization from experimental and theory data

I II III

Courtesy of Ying Yang, CompuThermhttp://www.computherm.com/pandat.html

First Ab Initio CALPHAD Lower Mantle Result

T[K]

x(FEO)

0

950

1900

2850

3800

0.0 0.2 0.4 0.6 0.8 1.0

x(FEO)

T[K]

MGO FEO

RS

iB8

iB8

RS+iB8

0 0.2 0.4 0.6 0.8 10

950

1900

2850

3800

T[K]

x(FEO)

0

950

1900

2850

3800

0.0 0.2 0.4 0.6 0.8 1.0

x(FEO)

T[K]

MGO FEO

RS

RS+iB8iB8

RS+iB8

RS iB8

0 0.2 0.4 0.6 0.8 10

950

1900

2850

3800

First steps completed More accurate expressions and fitting to experiment needed

P=50 Gpa P=100 Gpa

Conclusions

Identified key spin dependent interactions− Crossover pressure vs.

composition, structure

− Mg-Fe interaction vs. spin state

Constructed first-principles based thermodynamic model− Prediction of phase separation in

ferropericlase

Future challenges− Approach: LDA, GGA, +U, …

− Accuracy of models

− Full lower mantle thermodynamic model (multiphase, Fe2+/Fe3+, Al)

Ferropericlase

Perovskite

700

4000D

epth

(km

)

CFe

2-phase

iB8 HS

B1 mixed spin

HS

LS

1800

2900

0.25 0.750-MgO 1-FeO

Please see Amy Bengtson’s poster!

(Fe,Mg)O Very Complex …

Structural changes (B1, NiAs)

Jahn-Teller distortions Magnetic order Mg-Fe composition Metal-insulator

transition Spin transition Point defects

(vacancies, Fe3+) P,T

Lin, et al., PNAS '03

B1: Cubicparamagnetic

rB1: rhombantiferromagnetic

NiAs

(Mg,Fe)O phase stability

All couple together – “Perfect storm” alloy

DOS from Ab Initio

FeO AF rB1

-2

-2

-1

-1

0

1

1

2

2

-10 -5 0 5

E (eV)

States/cell

t2g spin-up

t2g spin-dn

eg spin-up

eg spin-dn

The Thermodynamic Terms

Pressure drives HS → LS (volume and crystal field effects)Temperature effects all stabilize HS or increase mixing

F to flip a spin = FHS – FLS =

– EHund + Exf + P(VHS-VLS) – TSconf+ Fmag + Fvib + Felec

P (GPa)0

–EHund

P(VHS-VLS)Exf

E F

HSLSFmag

Fvib

LS more stable

HS more stable

Felec

–TSconf

First-Principles Spin Transition Calculations

0 Persson - Private communication1 Cohen, et al. Science '972 Gramsch, et al. Am. Min. '033 Fang, et al. Phys Rev B '994 Badro, et al. Phys Rev Lett '995 Milner, et al. ‘046 Kondo, et al. J App Phys '007 Guo, et al. Phys. Cond. Matt. '028 Feng and Harrison, Phys Rev B '049 Eto, et al. Phys. Rev. B '0010 Parlinski, Eur Phys J B '0211 Sarkisyan, et al. JETP Lett. '0212 Rohrbach, et al. J Phys C '0313 Chattopadhyay, et al. J. Phys. Chem.

Solids '85; Physica '8614 Dufek, et al. Phys. Rev. Lett. '9515 Pasternak, et al. Phys. Rev. Lett. '9516 Rueff, et al. Phys. Rev. Lett. '9919 Nekrasov et al. Phys. Rev. B ’0320 Yan, et al. Phys. Rev. B ’04

System Ab Initio (GPa) Expt. (GPa)

FeO 200 (GGA)1

>250 (GGA+U)2,3

>1434

905

MnO 150 (GGA)1 906

CoO 90 (GGA)1 907

NiO 230 (GGA)1

>400 (GGA)8

>600 (B3LYP)8

>1419

FeBO3 23 (GGA)10

40 (GGA+U)0

4611

MnS2 0 (GGA)12

11 (GGA+U)12

1413

NiI2 25 (GGA) 14 19 (GGA) 15

FeS 6 (GGA+U)12 6.516

LaCoO3 140 K17 35-100 K18

0

40

80

120

160

200

0 0.2 0.4 0.6 0.8 1Fe Concentration

Pressure (GPa)

8.5

9

9.5

10

10.5

11

Volume (A

3 /atom)

Understanding PT vs. CFe TrendThe Mg Compression Argument

PT

▲Volume (P=0GPa)HS

LS

Mg compresses Fe-HS HS less stable PT ↓ Mg expands Fe-LS LS less stable PT ↑ Affect on PT unclear?

P=0: R(Fe-HS)>R(Mg)>R(Fe-LS)

Spin Transition Calculations Sensitivity: (Mg,Fe)SiO3

PT

200 Gpa

CFe = 12.5% CFe = 25%

170 Gpa

140 Gpa

dFe-Fe = 4.98 Ǻ

dFe-Fe = 3.38 Ǻ GGA-PW (Perdew, et al. PRB ’92)

GGA-PBE (Perdew, et al. PRL ’97)

Local order

Exchange-correlation parametrization

Make this cofig 210-170, Fe2+-Fe3+ 200-140, And summarize