simplest Morse : simplest (1 Well-chosen functional form is more useful than elaborate fitting strategies!! Phys. Rev. 34, 57 (1929 ( 2 E ij = D e 1" exp("#(r ij " r e 2 r ij = r i " r j r ij =r e E ij : D e =, α=, r e = 2
: simplest (2 Lennard-Jones i< j + $ E = 4" & # ' 1 - -% r i, j, ij ( 12 $ * # ' & % r ij ( 6. 0 0 / E r ij = 2 1/ 6 " ~ 1.1" "# Proceedings of the Physical Society 43, 461 (1931 (van der Waals ( /r 3 - -1/r 6 ( : Ar 84K 87K (
: Si (1 Si as a covalent system Stillinger-Weber Phys. Rev. B 31, 5262 (1985 E = i< j i< j<k " v 2 (r ij + " v 3 ( r ij, r i, j i, j,k ik for diamond structure v 2 deep well v 3 ( r 1, r 2 = g(r 1 g(r 2 h(l 12 l 12 = r 1 " r r 2 r 1 r 2 h(l = (l + 1 3 2 sp 3
: Si (2 Tersoff (T1,T2,T3 Phys. Rev. Lett. 56, 632 (1986; Phys. Rev. B 32, 6991 (1988; Phys. Rev. B 38, 9902 (1988 Si-Si Si i< j $ [ ] E = " R (r ij + p(# ij " A (r ij i, j " R (r ij repulsive, " A (r ij attractive p(" ij bond order " ij = # v * 3 ( r ij, r ik Stillinger-Weber k 4
: Si (3 EDIP for Si (environment-dependent interatomic potential M.Z. Bazant et al.,phys. Rev. B. 56, 8542 (1997 Tersoff 2 3 (= EDIP Si (= EDIP for C (environment-dependent interatomic potential N. Marks, J. Phys.: Condens. Matter 14, 2901 (2002 EDIP for Si C Tersoff diamond graphite
: 2 2 / e.g., Elastic moduli: i< j For E = " v 2 (r ij i, j C "#$% = 1 2&N i.k / i,k ( d 2 v 2 (r ik ' 1 dv 2 (r ik + r ik," r ik,# r ik,$ r ik,% * 2-2 dr ik r ik, r ik ( dr ik transferability
: ( AMBER
: (1 (sp-valent metals Nearly-free electron approach pseudopotential 2 : E = # S * ( q r S( q r $ V ion ps ( q r 2 %( q r 2&( q r r q "0 with $ S( q r = 1 exp("iq r # r j N j "( q r =1# 8$%( q r /q 2 "( r q =1# 8$%( r q [1# G(q]/q 2 "( q r = linear response func. (Lindhard = % r p,spin V ion ps ( q r = Fourier Transform of the pseudopot. f fermi (" r p + r q # f fermi (" r p " r p #" r p + r q + i$
: (2 E = E 0 + 1 N i< j #"(r ij "(r = 4Z 2 with ion #r i, j & % 0 V ion ps (q 2 $(q sin(qr q dq "(r Al 2k F 1/3 Φ cohesive energy cohesive energy LDA-DFT 2 Al Dragens et al., Phys.Rev. B 11, 2226 (1975
: -EAM (1 Embedded-atom method i< j #i E = #"(r ij + # F($ i with " i = $ " atom (r ij i, j i j "(r ij 2 ( F(" i ρ i ( : : F(" i = a" i = a $ " atom (r ij #i j 2 F(" i = a" i # b " i fitting LDA-DFT Puska et a., (1981
: -EAM (2 EAM 1 Φ ρ (r 0 E /N = AZ " B Z A = 1 2 "(r 0 + a# atom (r 0 (< 0 B = b " atom (r 0 (> 0 Z= E vs. Z E( Z 0 < E( Z, Z = Z 0 E/N 1980 embedded-atom method / Finnis-Sinclair N-body pot. / the glue model Z Al LDA-DFT results by Heine et al. (1991
: -EAM (3 EAM 2 vacancy formation energy ( B=0 EAM 2 "#U vac = 2A = Z " 2A # Z " A = ZA A=0 EAM "#U vac = Z=4 cohesive energy : minus U coh = "ZA (> 0 " #U vac = $ZA = U coh Z=4 cohesive energy : minus U coh = B Z (> 0 = "BZ( Z " Z "1 # " B 2 Z " #U vac = B 2 Z = 1 2 U coh Al "U vac ~ 0.5U coh EAM
: : ES+, ReaxFF Goddard group (Caltech
: (1 ES+ Metal/oxide V ES+ =V EAM +V ES V EAM = # F [" i ] + # $ ij r ij i { } V ES =! 0 i q i + 1 2 J 0 2 " i q i + 1 2 i i< j ( F.H. Streitz and J.W. Mintmire, Phys. Rev. B 50, 11996 (1994 Embedded-Atom method : " i% j $ "V ES+ ( # d 3 r 1 d 3 i ( r 1 ;q i # j r 2 ;q j r 2 $ "q i V ES (Σ i q i =0 {q i } r 12 O 2 on Al " i 1s
: (2 ES+ α-alumina Streitz-Mintmire Experiment E c (ev/al 2 O 3 31.8 31.8 a (Å 5.13 5.13 C 11 (GPa 537 497 C 12 (GPa 180 164 C 13 (GPa 106 111 C 33 (GPa 509 498 C 14 (GPa -30-24 C 44 (GPa 130 147 C 66 (GPa 179 167
: Al Al Nanoparticles --- (radius~100å In high O-density: explosion Application: Ignition powder In low O-density and room temp.: Unique oxidation behavior: amorphous surface oxides (20 ~ 40Å Application: Nanophase materials... unique electrical, chemical, & mechanical behavior Aumann et al. (1995
: (3 ( =1500K 1ps=10-12
: (4 70 Å