cluster expansions: treating the effects of strain gus l. w. hart northern arizona university...
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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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Average July high/low: 28° C (82° F) and 10° C (50° Average July high/low: 28° C (82° F) and 10° C (50° F)F)Average July relative humidity: 37% (afternoon)Average July relative humidity: 37% (afternoon)Average yearly snowfall: 2.7 m (108 in)Average yearly snowfall: 2.7 m (108 in)
Flagstaff is one of the 10 sunniest locations in the Flagstaff is one of the 10 sunniest locations in the US but...it is also one of the top 10 snowiest citiesUS but...it is also one of the top 10 snowiest cities
Northern Arizona UniversityNorthern Arizona UniversityAltitude: 2100 m (7000 ft)Altitude: 2100 m (7000 ft)
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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