Cluster Expansions: Cluster Expansions: Treating the effects Treating the effects

of strainof strain

Gus L. W. HartGus L. W. HartNorthern Arizona Northern Arizona

UniversityUniversityFlagstaff Arizona, USAFlagstaff Arizona, USA

Flagstaff Arizona

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Generalized Ising model

Formally exact but useless unless sums can be Formally exact but useless unless sums can be truncatedtruncatedFormally exact but useless unless sums can be Formally exact but useless unless sums can be truncatedtruncated

An Illustration:Cluster expansion

(Ising)

Approach:Approach:Determine Determine ZZ for a “few” structures via DFT for a “few” structures via DFTUse fitting to find values for Use fitting to find values for { J }{ J }Predict Predict ZZ for new structures, check against for new structures, check against

DFTDFTAdd new structures to your computed set Add new structures to your computed set

of of ZZ’s’sIterate until we have a Iterate until we have a universaluniversal set of set of JJ’s’s

Approach:Approach:Determine Determine ZZ for a “few” structures via DFT for a “few” structures via DFTUse fitting to find values for Use fitting to find values for { J }{ J }Predict Predict ZZ for new structures, check against for new structures, check against

DFTDFTAdd new structures to your computed set Add new structures to your computed set

of of ZZ’s’sIterate until we have a Iterate until we have a universaluniversal set of set of JJ’s’s

➙➙ Result: a robust, Result: a robust, extremely fastextremely fast, model , model

Hamiltonian Hamiltonian (millions of (millions of atoms, millions of configurations)atoms, millions of configurations)

➙➙ Result: a robust, Result: a robust, extremely fastextremely fast, model , model

Hamiltonian Hamiltonian (millions of (millions of atoms, millions of configurations)atoms, millions of configurations)

Cluster Expansion for

-

Coherent semi-infinite slabs

Cluster expansion predictsCluster expansion predictsvanishing formation enthalpyvanishing formation enthalpy

Who cares?Who cares?

Have a profound effect on theHave a profound effect on theproperties of metalsproperties of metals

Precipitates!

Huh?

What does that have to What does that have to do with coherent semi-do with coherent semi-

infinite slabs?infinite slabs?

Coherent semi-infinite slabs

Cluster expansion predictsCluster expansion predictsvanishing formation enthalpyvanishing formation enthalpy

Is that correct?Is that correct?

+Alloy of two mismatched

components

What happens at an (coherent) What happens at an (coherent) interface?interface?And the And the energy?energy?

The resulting strain energy is The resulting strain energy is “long-range.”“long-range.”

What do we mean by that?What do we mean by that?

How is it that the Ising model How is it that the Ising model (cluster expansion) can’t handle (cluster expansion) can’t handle this?this?

How can this be fixed?How can this be fixed?

Superlattices...•A superlattice is like small, semi-infinite

slabs(What the #$%@ does that mean?)

•Consider a 2n-layer superlattice AnBn

•What happens as n gets large?(Or as k gets small if we think in k-space?)

It’s even worse...

Strain energy is Strain energy is direction direction dependentdependent

Non-analyticity

In the limit of infinite-layers (k goes to zero) the In the limit of infinite-layers (k goes to zero) the strain reaches a finite value...strain reaches a finite value...

But that value depends on the orientation of But that value depends on the orientation of the interface!the interface!

I.e., the limit k→0 depends on the direction in I.e., the limit k→0 depends on the direction in which we take the limitwhich we take the limit

Account for the “long-range” part of the energy by adding

another term

Ising model handles this part

k-space strainmodel

In principle, any “correct” model for

the strain can be used...

Must have the correct non-analyticity

We can still parameterize ab-initio

These terms cancome from one-atom-cell DFT calculations

Epitaxial strains

Epitaxial strains

Depends on both concentration and in-plane lattice constant

We end up with an “upside-down parabola” for each direction

Stefan Mueller’s work

Stefan Mueller’s work