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

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)

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) 251-310) Nelson et al. PRL 61 (1988) 1977

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

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

SMALL CLUSTER GROWTH ON Pt(100) AGREES WITH EXPERIMENT P. Schwoebel, S.M. Foiles and G. Kellogg, Phys. Rev. B 40 (1989) 10639.

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

(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

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

EAM IS A POWERFUL TOOL FOR PREDICTING SURFACE RECONSTRUCTION 2 energies at 0 K surface relaxation has little effect

TEMPERATURE DEPENDENCE OF (110) PHASE TRANSITION AGREES WITH EXPERIMENT Calculated Experiment Pt 750 K 960 K Au 570 K 650 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.

THE EAM CANNOT BE USED FOR A WIDE CLASS OF MATERIALS Solid C12 / C44 - 1 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 -0.2 MEAM MoSi2 -0.5 MEAM Pu -0.2 MEAM Materials with C12 / C44 - 1 < 0 have a significant amount of directional bonding and cannot be described by EAM

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

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

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

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

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

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

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

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)

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., 1992. 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.”

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

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

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.

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

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

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