Automating first-principles phase diagram calculations

Axel van de Walle (Northwestern University)

Mark D. Asta

(Northwestern University)

Gerbrand Ceder (MIT)

This work was supported by: DOE under contract no. DE-F502-96ER 45571. NSF under program DMR-0080766 and DMR-0076097.

Composition-temperature phase diagrams

Thermodynamic properties of stable and metastable phases,

Short-range order in solid solutions,

Thermodynamic properties of planar defects

Morphology of precipitate microstructures

Previous successes of

first-principles thermodynamic

calculations

(Ducastelle (1991), Fontaine (1994), Zunger (1994,1997),

Ozolins et al. (1998), Wolverton et al. (2000),

Ceder et al. (2000), Asta et al. (2001).)

Automate the process in order to make

this tool available to a wider community

First-principles

Phase Diagram Calculation

Theory is well established:

(Ducastelle (1991), Fontaine (1994), Zunger (1994))

Our goal:

But: the task is tedious in practice.

Year

Tim

e

1980 2000

Computer

Human

E(σ1,...,σn)

The cluster expansion

Alloy system Lattice model

+1+1

+1

+1+1

+1+1+1

+1+1

+1

+1 +1

E(r1,...,rn)

ri

Parametrize configurational-dependence of the energy E

with a polynomial in the occupation variables σi:

E(σ1,...,σn) = Σ Jij σiσj + Σ Jijk σiσjσk + ...{i,j} {i,j,k}

= Σ Jα σααwhere

and

, ,... ..., , ,α =

i

∋

αΠ σiσ

α

αcluster+1

+1 +1

+1-1

-1 -1

-1

+1

+1= −1

-1

+1=

Structures Clusters

, ,... ..., , ,...,, , ,

Least Squares Fit

E(1) = Σα

Jα σα(1)

E(2) = Σα

Jα σα(2)

Jα

First-principles

Calculations

(2)(1) (3) (4)α1 α10α2 α11

Cluster expansion fit

Questions:

1) Which clusters α1, α2, ... to include?

2) Which structures (1), (2), ..., to include?

Strange ground state?

Decaying ECI?

Mean Square Error small?

Error on ground states small?

Have supercomputer time to burn?

Deadline coming up?

Tired already?

Look for the magic clusterFind bad structure and kill it

Force this structure not to be a ground state

Can I remove one structure

and predict its energy?

Add not-too-complicated cluster

Add a structure that makes things work better

Start with simple clusters and structures

Compute many many structural energies

Add simple structure and/or clusters

least-squares fit

Stop doing cluster expansions

Cluster expansions construction:

The Old Way

Yes

No

Yes

No

Yes

No

Yes

No

Yes

No

Yes

No

YesNo

Yes

No

Done!

Cluster expansion construction

MAPS

Monte Carlo Simulations

Emc2

(MIT Ab-initio

Phase Stability Code)

(Easy Monte Carlo Code)

Lattice geometry Ab-initio code parameters

Ab-initio code

Thermodynamic quantities Phase diagrams

Effective cluster interactions Ground states

"Predictors"

Elastic or

Electrostatic

Energy

ex:ex: VASP

Alloy Theoretic Automated Toolkit

12

3

Find best

cluster expansion

Filter

Filter

Job manager

geometry

energy

Find best

structureinput files

output files

"ready"

"error"

cluster

expansionground

states

Array of

processors

MAPS

Lattice geometry Ab-initio code parameters

Alloy theory

"Engine"

Ab-initio code

Interface

Processes

InterfaceHardware

Cluster expansion construction module

(MAPS)

Cross-Validation

score (CV)

Cross-Validation

score (CV)

Number of regressor

Cross-Validation

CV = Σ ( Ei − E-i )2

E E E

Example for polynomial fit

i=1

n

Energy of structure i

predicted from a fit to structures

1, 2, ... , i−1, ,i+1, ... , n

i removed

True energy of structure i

Squar

e er

ror

Mean Square

Error (MSE)

Mean Square

Error (MSE)

CV prevents under- or over- fitting

"Leave-one-out estimator"

Cluster Hierarchy

Example: square lattice

Rules:1) Include cluster only if all its subclusters are also included.

2) Include cluster only if all smaller clusters are also included.Note: α smaller than β iff

max ||ri − rj|| ≤ max ||ri − rj||i,j∈α i,j∈β

rirj

Include these clusters only if those clusters included.

0 20 40 60 80 100

En

erg

y

Concentration (%)

Incorrect ground state

predicted energypredicted energy ab-initio energy

Choose ECI with

cross-validation

Flag incorrectly

ordered structures

none ?

done

Increase their

weight in the fit

Ensuring ground state accuracy

(from LS fit)

Structure generation

0 20 40 60 80 100

Ener

gy

Concentration (%)

Criteria:

1) look for predicted ground states

2) Minimize variance

(look for noncoplanar structure in correlation space)

ρ1ρ2

ρ3

3 3 7.35 90 90 120

1 2 1.33

-2 -1 1.33

1 -1 1.33

0 0 1.0000 O

2 1 0.6666 Co

1 2 0.3333 O

0 0 0.0000 Li,Vac

Axes

Cell

Atoms

a b c α β γ

[INCAR]

PREC = high

ISMEAR = -1

SIGMA = 0.1

NSW=41

IBRION = 2

ISIF = 3

KPPRA = 1000

DOSTATIC

do static run

k-point density

(k point per reciprocal atom)

Standard VASP tokens

1) Lattice geometry

2) First-principles code parameters

Input Files

-450-400-350-300-250-200-150-100-50

0

0 20 40 60 80 100

Ener

gy (m

eV)

Concentration(at % Ti)

generated calculated

1) Formation Energies and Ground States

-40

-20

0

20

40

60

1 1.5 2 2.5 1 1.5

ECI (

meV

)

r/rnn:

pairs triplets

2) Effective Cluster Interactions

Example output (hcp Ti-Al system)

Energy Predictors

Presence of long range interactions:

pai

r E

CI

distance

Elastic interactions

Electrostatic interactions

Hard to fit so many ECI...

1) Look for analytic solution for long-range ECI behavior:

True ECITrue ECI

Analytic prediction

2) Subtract out long-range analytic contribution

Energy predictors easy to add to MAPS:

No change needed in the code,

just link with user-specified routines

3) Fit short-range contribution to ab initio calculations

Solution:

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.2 0.4 0.6 0.8 1

1010 0

110110

111111

210210

-0.08

0

0.08

-0.08

0

0.08

-0.08

0

0.08

-0.08

0

0.08

-0.08

0

0.08

-0.08

0

0.08

-0.040

0.04

-0.04

0

0.04

-0.04

0

0.04

Const

ituen

t st

rain

ener

gy (

eV/a

tom

)Angular-dependence of the

k-space expansion coefficients

Concentration (at % Au)Cu Au

Constituent strain formalism implementation

Concentration-dependence of the

constituent strain energy

Input:

Output:

100

110

111

210

Superlattice

Directions

ab-initio

code

parameters

(Laks, Ferreira, Froyen, and Zunger, 1992)

+

Monte Carlo Simulations

E = Σα Jα σα

From cluster expansions:

to thermodynamic quantities:

Phase diagrams:

Free energies:

F

concentration

Tasks to perform:

How to determine when target accuracy reached?

How to detect phase transitions?

How to efficiently calculate phase boundaries?

Equilibration and Averaging times

Equilibration time Averaging time

Equilibrium not reached at t1

Equilibrium reached at t2

Given tolerance: ε

> ε

< ε

Choose averaging time such that

(E, x,E, x, SRO) SRO)on some thermodynamic quantity Q =

Detecting phase transitions

2) Use cross-validation to find optimal order of polynomial

(Accounting for noise in MC averages)

1) Fit polynomial through (Qi, Ci)

4) Perform statistical test of equality between Qn+1 and Qn+1

3) Extrapolate Qn+1 using fitted polynomial

Q(C)

Thermodynamic

quantity

Control

variable

Goal: find discontinuity in

(E, x,E, x, SRO) SRO) (µ, µ, T)

µ

φ

β

β + dβ

dµ E − E µ

dβ β(x − x ) β=

γ

αγ

α

γ

α

Phase boundary tracing

µ(β)µ(β + dβ)

as temperature β changes,

need to update µ to preserve φγ = φαPrinciple:

µ(β) is known find xγ(µ(β)) and xα(µ(β))

x

γ α

-gs=3

-mu0=3.5

-mu1=4.5

-dmu=0.05

Unit cell

Simulation Cell

enclosed

sphere

radius

0 0.2 0.4 0.6 0.8 1

Concentration

0

1

2

3

4

5

µ=1

µ=2

µ=3µ=4

µ=5

µ = 3.5µ = 3.5µ = 4.5µ = 4.5

-er=35

-dx=1e-3

-T0=100

-T1=1000

-dT=20

-k=8.617e-5

Temperature range

Boltzman's kB

-dx=1e-3

-er=35

-dT=20

-k=8.617e-5

Target precision

Thermodynamic

Integration

Phase

Boundaries

Determination

Input parameters

(Monte Carlo Code)

-gs1=3

-gs2=4

E

Free energy surfaces

CaO-MgO system

Ti-Al system

Emc2 output:

Emc2 ouput:Phase diagrams

Short-range order calculations

Calculated diffuse X-ray scattering in Ti-Al hcp solid-solution

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

05

101520253035404550

0 1 2 3 4 5 6

γk (

mJ/

m2)

k

T/Tc =0.60xAl =6%xAl =8%xAl =10%xAl =12%

Energy cost of creating a diffuse anti-phase boundary

in a Ti-Al short-range ordered alloy by sliding k dislocations