The fundamental laws necessary for the mathematical

Report
First-Principles Study of Fe Spin
Crossover in the Lower Mantle
Dane Morgan, Amelia Bengtson
Materials Science and Engineering
University of Wisconsin – Madison
Second VLab Workshop
University of Minnesota
August 5-10, 2007
Computational Materials Group
University of Wisconsin - Madison
http://matmodel.engr.wisc.edu/
 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
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
High spin
M = 4mB
Intermediate spin
M = 2mB
Minority
Exf
eg
EHund
t2g
Majority
Low spin
M = 0mB
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
(Mg0.92Fe0.09)Si1.00O3
(Mg0.87Fe0.09)(Si0.94Al0.10)O3
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
First-Principles Calculations
Composition and Structure
(e.g., Mg0.75Fe0.25O)
Quantum mechanics
(+ approximations)
• Energies: Stability, Atomic Positions, …
• Electronic Structure: Spin state, Bands, …
• Additional modeling for T>0, optical properties, …
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, postperovskite)
− 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 firstprinciples + modeling
Calculating Spin-Transitions
E
LS
HS
DH=HHS–HLS
LS
PT
HS
VHS
VLS
V
P
First-Principles Prediction – (Mg,Fe)O
CFe = 19%, Theory, 2006
Tsuchiya, et al., Phys. Rev. Lett. ‘06
CFe = 25%, Expt, 2007
Lin, et al., Science TBP
First-Principles Fe-Spin Results
(apologies to those I missed!)
 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)
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, …
−
−
−
−
−
−
Composition: global and local chemical order
Valence
Site occupancy
Temperature
Structural relaxation
Magnetism
 Correct materials system parameters
Spin Transition Calculations Sensitivity:
Calculation Parameters - (Mg0.75Fe0.25)SiO3
PT
200 GPa
GGA
GGA-PW
(Perdew, et al. PRB ’92)
150 GPa
GGA-PBE
(Perdew, et al. PRL ’97)
Exchange-correlation effects
100 GPa
LDA
Sensitivity to calculation method - which is best?
Spin Transition Calculations Sensitivity:
Materials Parameters - (Mg0.75Fe0.25)SiO3
PT
200 GPa
dFe-Fe = 4.98 Ǻ
Fe2+
170 GPa
dFe-Fe = 3.38 Ǻ
GGA-PBE
Fe local order
140 GPa
(Perdew, et al. PRL ’97)
Valence effect
Al local order
Fe3+ + Al
Sensitivity to valence/configurations – need to compare like configurations
Spin Transition Calculations Sensitivity:
Materials Parameters - FeSiO3
PT
900 GPa
Cubic symmetry
240 GPa
MgSiO3 symmetry
(Cohen, et al. Science ’92)
(Stackhouse, et al. EPSL ’07)
Structural relaxations
77 GPa
No symmetry
(Bengtson, et al. Submitted)
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
Persson, et al., GRL ‘06
 Strong composition
-spin crossover
coupling
 What drives the
crossover?
 What drives
composition effect?
Ferropericlase: What Drives the Crossover?
Spin crossover (T=0) when
DH = EHS-ELS + P(VHS-VLS) = 0
 DE does not go to zero!
P∆V
∆E
 PDV term is the most
important driver of the
transition!
 Both DE, PDV terms
drive up crossover
pressure with Fe content
 Effect of chemical
pressure?
Understanding PT vs. CFe Trend
Chemical Pressure
7.8
 PT
160
▲Volume (P=100GPa)
7.6
HS
120
7.4
80
7.2
LS
40
7
0
6.8
0
0.2
0.4
0.6
Fe Concentration
0.8
1
P=0: R(Fe-HS)>R(Mg)≈R(Fe-LS)
 Mg compresses Fe-HS  HS less stable  PT↓
 Mg does not expand Fe-LS  LS unaffected  PT↔
 Increasing Mg pushes PT↓
Volume (A3/atom)
Pressure (GPa)
200
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?
 PDV still very important in
transition
P∆V
∆E
 DE terms drive down
crossover pressure with Fe
content
 Changes in DE due to
structural relaxations
(crossover pressure =
~900 GPa w/o relaxation!)
Crossover Pressure vs. Fe Composition
Strong Structural Coupling
Perovskite
Ferropericlase
 Transitions driven significantly by PDV terms
 Opposite trends due to structural relaxation in 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 (PDV)
makes a major contribution
to the spin crossover
energetics.
Perovskite
Ferropericlase
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
Thermodynamic
Modeling
F  P, T ,{c}  UT 0  PV  Fconf  Fmag  Fvib  Felec
CALPHAD
Phase stability, Fe partitioning,
Fe spin states, Densities, …
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
High Spin
Tc(100GPa)≈900K
Low Spin
Tc(100GPa)≈4500K
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
First-Principles study of Fe Spin Crossover
T>0
First-Principles Model for Ferropericlase
MgO
FeO-HS
FeO-LS
 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
Free Energy Model
F  P, T ,{c}  U T 0  Fconf  Fmag  Fvib  Felec  PV
First-principles
 U dis  PV  TSconf  Fmag  Fvib  Felec
U dis  First-Principles energies, SQS to simulate disorder 1
  HS ln cFe
  HS  cFe
  LS ln cFe
  LS  
TSconf  kT cMg ln cMg  cFe ln cFe  cFe  cFe
Fmag  cFe  HS kT ln  5 
First-principles
13
 3 
Fvib  -3kT  4 3  log T TD   ; TD  0.617 

 4 
Felec  cFe  HS kT ln  3 (3 degenerate t 2g states) 3
1.
2.
3.
12
h  B


kB  M 
13
2
S. H. Wei, et al., Phys. Rev. B, '90
A. van de Walle and G. Ceder, Reviews of Modern Physics, '02
G. R. Burns, Minerological Applications of Crystal Field Theory, '93
Fitting The Free Energy
F  P, T ,{c}  Udis  PV  Fvib  TSconf  Fmag  Felec
Fit and interpolate
MgO
FeO-HS
Analytic
 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
FeO-LS
F   Fc
i i   Fij ci c j 
i
i j
F
i  j k
cc c
ijk i j k
Fitting The Free Energy
MgO
 Fitting grid for B1 (NaCl)
− Mixed spin data uncertain
so assume no Fe-HS – FeLS interaction
FeO-LS
FeO-HS
MgO
 Fitting grid for iB8 (i-NiAs)
− Ab initio  almost no LS
− Ab initio  almost no Mg
solubility
 Easy to fit!
FeO-HS
FeO-LS
Phase Diagram
HS
1800
B1 mixed spin
2900
P≈140GPa
iB8 HS
2-phase
LS
4000
0-MgO
0.25
CFe
0.75
1-FeO
CFe-HS/CFe
Depth (km)
700
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
http://www.computherm.com/pandat.html
II
III
Courtesy of Ying Yang, CompuTherm
RS
First Ab Initio CALPHAD
Lower Mantle
Result
iB8
iB8
RS+iB8
P=50 Gpa
3800
3800
P=100 Gpa
3800
3800
iB8
2850
2850
iB8
RS
T[K]
T[K]
T[K]
T[K]
2850
2850
1900
1900
RS
1900
1900
RS+iB8
950
950
950
950
RS+iB8
00 0
0.0
MGO
0.2
0.2
0.4
0.4
0.6
0.6
x(FEO)
x(FEO)
0.8
0.8
1
1.0
00 0
0.0
FEO
MGO
0.2
0.2
0.4
0.4
0.6
0.6
x(FEO)
x(FEO)
 First steps completed
 More accurate expressions and fitting to experiment needed
0.8
0.8
1
1.0
FEO
Conclusions
Perovskite
 Identified key spin dependent
interactions
− Crossover pressure vs.
composition, structure
− Mg-Fe interaction vs. spin state
Ferropericlase
 Constructed first-principles based
thermodynamic model
− Prediction of phase separation in
ferropericlase
700
− Approach: LDA, GGA, +U, …
− Accuracy of models
− Full lower mantle thermodynamic
model (multiphase, Fe2+/Fe3+, Al)
Depth (km)
 Future challenges
HS
1800
B1 mixed spin
iB8 HS
2900
4000 LS
0-MgO
2-phase
0.25
CFe
Please see Amy Bengtson’s poster!
0.75
1-FeO
(Fe,Mg)O Very Complex …
 Structural changes
(B1, NiAs)
 Jahn-Teller distortions
 Magnetic order

Mg-Fe
composition
B1: Cubic
 Metal-insulator
paramagnetic
NiAs
transition
rB1: rhomb
 Spin transition
antiferromagnetic
 Point defects
3+)
(vacancies,
Fe
Lin, et al., PNAS '03
 P,T
All couple together – “Perfect storm” alloy
(Mg,Fe)O phase stability
DOS from Ab Initio
FeO AF rB1
2
2
States/cell
1
1
0
-10
-5
-1
t2g spin-up
-1
t2g spin-dn
-2
eg spin-up
-2
eg spin-dn
E (eV)
0
5
The Thermodynamic Terms
DF to flip a spin = FHS – FLS =
– EHund + Exf + P(VHS-VLS) – TDSconf+ DFmag + DFvib + DFelec
LS more
stable
Exf
E
P(VHS-VLS)
–TDSconf
0
HSLS
HS more
stable
DF
DFvib P (GPa)
DFelec
DFmag
–EHund
Pressure drives HS → LS (volume and crystal field effects)
Temperature effects all stabilize HS or increase mixing
First-Principles Spin Transition Calculations
System
Ab Initio (GPa)
Expt. (GPa)
FeO
200 (GGA)1
>1434
>250 (GGA+U)2,3 905
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 Persson - Private communication
1 Cohen, et al. Science '97
2 Gramsch, et al. Am. Min. '03
3 Fang, et al. Phys Rev B '99
4 Badro, et al. Phys Rev Lett '99
5 Milner, et al. ‘04
6 Kondo, et al. J App Phys '00
7 Guo, et al. Phys. Cond. Matt. '02
8 Feng and Harrison, Phys Rev B '04
9 Eto, et al. Phys. Rev. B '00
10 Parlinski, Eur Phys J B '02
11 Sarkisyan, et al. JETP Lett. '02
12 Rohrbach, et al. J Phys C '03
13 Chattopadhyay, et al. J. Phys. Chem.
Solids '85; Physica '86
14 Dufek, et al. Phys. Rev. Lett. '95
15 Pasternak, et al. Phys. Rev. Lett. '95
16 Rueff, et al. Phys. Rev. Lett. '99
19 Nekrasov et al. Phys. Rev. B ’03
20 Yan, et al. Phys. Rev. B ’04
Understanding PT vs. CFe Trend
The Mg Compression Argument
11
 PT
▲Volume (P=0GPa)
160
10.5
HS
120
80
LS
40
0
10
9.5
9
8.5
0
0.2
0.4
0.6
0.8
1
Fe Concentration
P=0: R(Fe-HS)>R(Mg)>R(Fe-LS)
 Mg compresses Fe-HS  HS less stable  PT ↓
 Mg expands Fe-LS  LS less stable  PT ↑
 Affect on PT unclear?
Volume (A3/atom)
Pressure (GPa)
200
Spin Transition Calculations Sensitivity:
(Mg,Fe)SiO3
PT
200 Gpa
170 Gpa
CFe = 12.5%
CFe = 25%
dFe-Fe = 4.98 Ǻ
Make this cofig 210-170, Fe2+-Fe3+ 200-140
And summarize
dFe-Fe = 3.38 Ǻ
GGA-PW
(Perdew, et al. PRB ’92)
Local order
140 Gpa
GGA-PBE
(Perdew, et al. PRL ’97)
Exchange-correlation
parametrization

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