1. LECTURE 2 M. I. Baskes Mississippi State University
University of California, San Diego
and
Los Alamos National Laboratory

2. COURSE OUTLINE Some Concepts Dislocation Motion Bond energy CREATOR
Many body effects
DYNAMO
Transferability
XATOMS
Reference state
Common Neighbor Analysis (CNA)
Screening
Models
Pair potentials
Embedded Atom Method (EAM)
theory
examples
Modified Embedded Atom Method (MEAM)

3. BULK PHONONS AGREE WITH EXPERIMENT
Phonon frequencies for Cu calculated from the dynamical matrix (see M. S. Daw, S. M. Foiles, and M. I. Baskes, Materials Science Reports 9 (1993) )
Nelson et al. PRL 61 (1988) 1977

4. BONDING OF SURFACE ATOMS IS QUITE DIFFERENT FROM BULK BONDING
15% softening of intralayer force constants
15% stiffening of interlayer force constants
Experimental Rayleigh modes
Phonon frequencies on relaxed Cu (100)
Nelson et al. PRL 61 (1988) 1977

5. EAM SUCCESSFULLY PREDICTS INITIAL NUCLEATION STAGES OF METAL OVERLAYER GROWTH
Field ion microscopy (FIM) used to investigate geometry of Pt clusters on Pt(100)
EAM used to simulate same geometries
Perfect agreement of predicted geometries
Atomic relaxations critical

6. SMALL CLUSTER GROWTH ON Pt(100) AGREES WITH EXPERIMENT
P. Schwoebel, S.M. Foiles and G. Kellogg, Phys. Rev. B 40 (1989)

7. PREFERRED GEOMETRY DEPENDS ON RELAXATION
Size
Stable geometry
Energy (meV) of high relative to stable
energy structure geometry
unrelaxed
relaxed 3
linear -1
159 4
island
438 52 5
-462 32 6
915
240

8. (110) SURFACE RECONSTRUCTION
Both Au and Pt (110) surfaces reconstruct to a structure with a (2x1) symmetry
A number of structures were originally proposed, but the “missing row” is now accepted as the correct structure
At high temperatures a transformation to a (1x1) structure occurs

9. A NUMBER OF FCC METALS UNDERGO A 2X1 MISSING ROW RECONSTRUCTION

10. EAM IS A POWERFUL TOOL FOR PREDICTING SURFACE RECONSTRUCTION2
energies at 0 K
surface relaxation has little effect

11. TEMPERATURE DEPENDENCE OF (110) PHASE TRANSITION AGREES WITH EXPERIMENT
Calculated Experiment
Pt K K
Au K K
Monte Carlo calculations (0,1/2) structure factor
only bulk properties used as input
first quantitatively realistic prediction of order-disorder surface phase transformation
M.S. Daw and S.M. Foiles, Phys. Rev. Letters 59 (1987) 2756.

12. THE EAM CANNOT BE USED FOR A WIDE CLASS OF MATERIALS
Solid C12 / C Appropriate Model
Ar 0.1 pair potential
Ni 0.2 pair potential
Mo 0.5 EAM
Cu 0.6 EAM
Au 2.7 EAM
Si MEAM
MoSi MEAM
Pu -0.2 MEAM
Materials with C12 / C < 0 have a significant amount of directional bonding and cannot be described by EAM

13. COMPLEX MATERIALS REQUIRE THE ADDITION OF ANGULAR FORCES
EAM uses a linear superposition of spherically averaged electron densities
MEAM allows the background electron density to depend on the local geometry  q θ

14. AN EQUILALENT FORM EXISTS FOR THE PARTIAL ELECTRON DENSITIES
Computationally more efficient (for short range)
Spatial moments of the electron densities
First sum over neighbors
Second sum over directions
EAM

15. PARTIAL ELECTRON DENSITIES REPRESENT DIFFERENT SYMMETRIES
s-symmetry
volume
l=1
p-symmetry
mirror plane
vacancy
l=2
d-symmetry
shear
l=3
f-symmetry
center of inversion
hcp/fcc
Barend Thijsse, Delft U. has related the partial electron densities to atomic environment type (AET) of Daams and Villars
AET’s form the stable solution points of natures Schrödinger equation
The number of AETs in nature is limited
If our potential interpolates accurately between sufficient AET’s, it will capture nature well

16. MODIFIED EMBEDDED ATOM METHOD (MEAM)
Universal Binding Energy Relationship
UBER
Background Electron Density
3-4 7
Embedding Function
Pair Potential 1
11-12 parameters

17. MODEL: MEAM Accuracy Computation Transferable Analytic or tabular
Volume
Coordination
Defects/strain
Computation
Analytic or tabular
Scales with number of atoms
Parallel architecture
Environmental dependence of bonding
Angular screening
Assumed functional forms
embedding function
electron density
background electron density
screening

18. AD-HOC ASSUMPTIONS ARE MADE
Angular dependence manifests itself in background electron density
Angular terms represented by partial electron densities as formulated
Background electron density given by sum of squares of partial electron densities
Form for embedding function
consistent with Pauling’s C bond length / bond order correlation
Exponential form for electron density
sum of exponentials looks like exponential over limited range
Only 1NN interactions in reference structure
BJ Lee, et al., 2000 generalized to 2NN

19. CURRENTLY DEVELOPED MEAM FUNCTIONS COVER MOST OF THE PERIODIC TABLE
BCC Li Be B C N O Na Mg Al Si P S Ar
FCC K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Sr Y Zr Nb Mo Ru Rh Pd Ag Cd In Sn Sb Ba La Hf Ta W Re Ir Pt Au Hg Tl Pb Bi Pr Nd
HCP Gd Tb Dy Ho Er
DIA CUB Th U Pu
Impurities
No functions

20. WHY ANGULAR (RATHER THAN RADIAL) SCREENING ?
Screening “fixes” the linear superposition of atomic densities
Moving an atom between two fixed atoms reduces their contribution of electrons to each other
This many-body screening effect is not handled by a radial cutoff (screening)

21. A FEW EXAMPLES OF TRANSFERABILITY AND ACCURACY
High pressure phase: Si, Badis, et al., 2003
“Our predicted phase transition pressure from diamond to bc8 is found to be 15.8 GPa in good agreement with values of 11 GPa given by Balnane et al., Therefore, we may conclude that a simple MEAM potential can reproduce with a fairly good approximation the high pressure phases of Silicon”.
Surface defects: Cu, Ag, Ni, Devytko, et al., 2000
“. . . calculated the formation energies of the isolated vacancy and the vacancy-adatom pair. The results predict the prevailing formation of pairs on the surfaces (110) and (100). With good accuracy, the calculated energy values coincide with those obtained from experiments”.
Surface reconstruction: Si(111), Takahashi, et al., 1999
“It is clearly shown that the 7 X 7 DAS structure is most stable in these NXN DAS structures, and 5 X 5 is more stable than 9 X 9. These results correspond to our experimental knowledge. This report suggests that the MEAM is a useful tool also for surface problems.”

22. MEAM FOR MULTI-COMPONENTS IS STRAIGHTFORWARD
Chose reference structure
Universal EOS for reference structure
cohesive energy
lattice constant
bulk modulus
(pressure derivative of bulk modulus)
Electron density scaling
shear elastic constants
4-5 parameters for each binary
Easy to get parameters from first principles calculation

23. A NUMBER OF MULTI-COMPONENT MEAM POTENTIALS EXIST
Binaries
Ni + Si, Al, Zr, Pt, H
Si + Mo, Ni, Ge, Al, Ar, C, B, H, O
Zr + Ni, N, O, H
Al+ Au, Ni, Ca, Cu, Si, O, Ti, V, Zr, Mn
Mg + Sc, Y, Pr, Nd, Gd, Tb, Dy, Ho, Er
Ca + Al, Cu
Pu + Ga, He
Cu + Au, Co, Sn
O + Fe, Cr, Ni
Fe + Ni, Cr, Pt
Am + N
Fe +He
Ternaries
Zr/O/H
Ca/Al/Cu
Ti/Al/Nb
Fe/Cr/O
Fe/Ni/O
Al/Mn/Ti
Al/Mn/V
Molecular
Pt/CO

24. A FEW EXAMPLES OF TRANSFERABILITY AND ACCURACY
Surface segregation: Pt80Fe20(111), Creemers, et al., 1996
“Low-energy ion scattering (LEIS) was used to study the surface composition of a Pt80Fe20(111) alloy surface as a function of temperature Only when the Monte Carlo method is combined with the more refined (modified) embedded atom method (MEAM) for energy description of an alloy is satisfactory agreement with experiment obtained.”
Thermodynamic properties: Mg/RE, Hu, et al., 2000
“The thermodynamic properties, such as the dilute-limit heats of solution, enthalpies of formation of disordered solid solutions and intermetallic compounds, for the binary Mg-RE alloys are calculated The obtained results are in good agreement with the available experimental data and with the results calculated using Miedema theory”.
Order-disorder transformation: Cu3Au, Tadaki, et al., 1997
“. . . the difference in ground state energy between perfectly ordered and disordered states becomes less in nano particles than in the bulk, and that the critical temperature T-c decreases markedly with decreasing particle size These experimental results appear to be accounted for by the present calculated ones qualitatively.

25. PREDICTED Mo/Si PHASE STABILITY IS IN CLOSE AGREEMENT WITH EXPERIMENTAL PHASE DIAGRAM
-0.5
-0.4
-0.3
-0.2
-0.1
-0.0
0.1
0.2
Formation Energy (eV)
0.25
0.5
0.75 1
Fraction Mo
Experiment
Predicted Metastable Phases
Predicted Stable Phases
M. I. Baskes, Mater. Sci. Eng., 1999

26. MEAM POTENTIALS ARE TRANSFERRABLE AND ACCURATE
Pu/Ga system
Enthalpy relative to pure elements
T = 0 K; P = 0
Many crystal structures investigated
C32 phase is predicted but not observed
Excellent agreement with experiment
ΔH (eV/atom)
MIB et at., JOM, 2003

27. TIMING FOR MEAM CALCULATIONS
Number of atoms
Number of processors
Processor speed (GHz)
Computer code
Computer time (ms)
/ atoms / time steps
X processor speed (GHz)
X number processors
256 1
2.4
DYNAMO
950.
455.*
3448
703.
57,944
524.
256,000 64
2.5
PARADYN**
119.
2,048,000
104.
PARADYN
31.***
* no angular screening
** modified for MEAM
*** EAM
PARADYN is the same as LAMMPS