Ab initio derived modified embeddedatom potentials for atomic processes at surfaces

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Ab initio derived modified embedded-atom
potentials for atomic processes at surfaces

• Youngho Shin and Matthias Scheffler
• Fritz-Haber-Institut der Max-Planck-Gesellschaft,
  Berlin, Germany
• Byung Deok Yu
• Dept. of Physics, University of Seoul, Seoul,
• Korea

(M)EAM Workshop at Eindhoven 23/24, Oct. 2002
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From ab initio to (semi) empirical

• Quantum calculation
• First principles
• Reliability proven within the approximations
• Basis sets,
• functional,
• all-electron or pseudo- potential ..
• Computationally expensive

• Based on fitting parameters
• Two body , three body, multi-body
  potential
• No theoretical background empirical
• Applicability to large system
• no self consistency loop and no eigenvalue
  computation
• Reliability ?

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Modified Embedded-Atom Method

• Ab initio and MEAM
• Quite good for bulk properties but poor
  agreement for surface properties. Why?
• Input data inconsistency using experimental set ?
  
• System specific data ?
• Fitting procedure ?

M. I. Baskes, Phys. Rev. B 46, 2727 (1992) M. I.
Baskes, Matter. Chem. Phys. 50, 152 (1997)
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MEAM formalism

• Total Energy density functional
• Embedding Energy
• Pair Potential from
• Universal Equation of State

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MEAM formalism

• Embedding Energy

with Angular dependent atomic density
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MEAM formalism

• Pair Potential

Interaction range radial cutoff limits
interaction range within the sphere radius r from
the atom and many-body screening
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MEAM formalism

• Many-body Screening with fixed larger range of
  radial cutoff
• In FCC,
• C3 (1st n.n.)
• C1( 2nd n.n)
• C1/3 ( 3rd n.n)

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Parameterization Procedure
Equation for the k system in n-dimensional
parameter space

• Baskes scheme
• Experimental data set
• Reference structure energy from the universal
  equation of state
• Bulk specific properties
• shear elastic constants, bulk vacancy formation
  energy
• structural energy difference (BCC FCC HCP)
• Analytic relation btw. parameter and data for the
  first n.n interaction potential

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Parameterization Procedure

• This work system specific scheme
• Theoretical data set from ab initio calc.
• Reference structure energy from
• ab initio data fitting
• Surface specific properties
• cohesive energy and bulk modulus,
• surface energies (001) (011) (111) , surface
  vacancy formation energy,
• adsorption energy (atop and hollow),
• diffusion barrier (hopping and exchange)
• Simultaneous parameters optimization using
  simulated annealing method (including screen
  parameter )

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Parameterization Procedure Baskes scheme (1st
n.n MEAM)

• Structure energy difference btw. BCC and FCC
• Shear elastic constants

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Parameterization Procedure Baskes scheme (1st
n.n MEAM)

• Structure energy difference btw. FCC and HCP for
  the same lattice constant and ideal c/a ratio
• Un-relaxed vacancy formation energy
• Effective parameters
• Fixed values

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System specific scheme atomic processes on (001)
surface

• Surface specific systems energy and its relaxed
  geometry obtained from ab initio calc. as input
  data

• (001),(011),(111) surface energy
• surface vacancy on (001) surface
• adsorption energy hollow and atop position on
  (001) surface
• hopping and exchange diffusion barriers on (001)
  surface

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System specific scheme atomic processes on(001)
surface
Preliminary setup Ab initio calculations Cu, Co

• Method DMol3
• Localized numerical atomic orbital basis
• Non-relativistic all electron potential with
  GGA(pbe) functional
• Tetrahedron method for the k-points integration

• Bulk properties
• zero temperature equations of state (bulk
  modulus, lattice constant, cohesive energy)
• structural energy difference (FCC,HCP,BCC)
• two shear elastic constants in FCC structure

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System specific scheme atomic processes on(001)
surface
Preliminary setup Ab initio calculations Cu, Co
Input data for the parameterization Flat surface
properties

• (001),(011),(111),surface energy from the at
  least 7-lyrs slab
• a surface vacancy, adsorption on hollow and atop,
  adatoms diffusion by hopping and exchange on the
  surface with p(3x3)-4 lyrs slab

External data set for test MEAM fidelity

• Cluster on the flat surface
• dimer, tetramer adsorption on the surface

• Stepped surface diffusion from (511)-3layrs
  vicinal surface
• diffusion along the edge and dissociation from
  the step
• overedge via hopping and exchange

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Preliminary setup equation of state for the
reference structure
System specific scheme atomic processes on(001)
surface

• Equation of State fitting from Binding Energy
  with homogeneous volume change (-30 30)

Cutoff
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System specific scheme atomic processes on (001)
surface

• Simulated Annealing method for minimizing errors
  between given data set and MEAM outputs

For n-input system given with its relaxed
geometry, MEAM calculates energy and maximum
force inside the system by varying parameter
vector P . By Metropolis algorithm, the most
promising area for optimization is focused.
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• All 9-parameters including screen parameter
  optimized

• Initial run with parameter ranges,
• then reduced them with condition,

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Initial run Input output systems energy
error within the condition,
Energy shows lower error in the 2nd n.n
interaction sets
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Initial run Atomic force maximum inside the
system geometry given by ab initio
optimization within the condition,
Atop system, lowest coordination system,
shows difficulty to satisfy the given ab initio
geometry
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The second run the optimized parameters set is
determined within the given ranges from the first
run
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Interaction range and parameters sets

• Cu0 Fit to the bulk properties using ab initio
  data
• Cu1 Fixed Cmin2.0 first nearest neighbor
  interaction only
• Cu2 Fixed Cmin 1.0 screened 2nd nearest
  neighbor interaction
• Cu3 Cmin also optimized promising candidate
• Co Cmin also optimized

Second n.n. interaction
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Results

• Input output data comparison Cu

• Poor quality
• bulk parameters set
• 1st n.n interaction parameters set

Low estimation for the dissociation energy
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Results

• Input output data comparison Co

geometry
Optimized MEAM shows good agreement in energy
and relaxed geometry with its DFT counterpart.
energy
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Results

• MEAM and DFT comparison for the outer data set

• Diffusion on the (115)
• vicinal surface Cu
• overedge by hopping
• and exhange,
• edge following,
• dissociation, and
• terrace diffusion barrier

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Results

• Comparisons with references Cu

Experimental diffusion barrier 0.30.4
eV depending on the experimental method and the
prefactor in fitting to Arrhenius law
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Summary and Conclusions

• Using ab initio input rather than experimental
  one
• conserves data consistency and enlarges available
  data
• All in one parameterization with simulated
  annealing method
• parameters optimized to reproduce energy and
  geometry
• independent of initial starting parameters
• Surface is quite different from bulk
• surface specific MEAM potential generated and
  compared with bulk specific MEAM potential
• Potential interaction range is scrutinized
• 1st nearest neighbor interaction is not enough
• Many-body screening factor has its own role Cu
  2nd n.n Co 3rd n.n
• Solid-liquid phase transition is rather explained
  well in surface specific parameters set