Molecular Dynamics simulations

Lecture 07:

Interaction models for metals

Dr. Olli PakarinenUniversity of Helsinki

Fall 2012

Lecture notes based on notes by Dr. Jani Kotakoski, 2010

METALS

I By definition, a metal is a material which is kept together bymetallic bonding

I Metallic bonding is a result of nearly-free electrons of a group ofatoms which bind with the positively charged atomic cores

I Of all elements, 80% are metals [image from A. Kuronen]

CRYSTAL STRUCTURES FOR METALS

I The most typical crystal structure for a metal is the hexagonalclose packed (HCP)

I The next most common isface centered cubic (FCC),whereas the third one is thebody centered cubic (BCC)

I With potentials describingthese three lattices, already∼ 60% of all stablesingle-element structures arecovered

I Both HCP and FCC have a packing density of 0.74 (filling ofspace with hard spheres)

I No structure, regular or not, has a denser packing

HCPI Hexagonal close packed

lattice vectors are:a = (a, 0, 0) α = β = 90◦

b = (a/2,√

32 a, 0) γ = 120◦

c = (0, 0, c)I The two atoms in the primitive

cell are located at:(0, 0, 0) & ( 2

3 , 13 , 1

2 )

I Atom in the HCP structure has 12 nearest neighborsI If the binding energy would depend only on the nearest-neighbor

distance (i.e., a pair potential), there would be no difference inthe energy of HCP and FCC

I Examples of materials with HCP lattice: He, Be, Mg, Ti, Zn, Cd,Co, Y, Zr, Gd and Lu

FCCI FCC’s lattice vectors are, of

course, cubic:a = (a, 0, 0)b = (0, a, 0)c = (0, 0, a)

I The atoms are located at:(0, 0, 0), (0, 1

2 , 0)( 1

2 , 12 , 0), (0, 1

2 , 12 )

I Again, there are 12 nearestneighbors and a 0.74 packing

I FCC metals tend to be soft and ductile over a wide temperaturerange

I Examples include: Au, Ag, Cu, Al and γ-Fe

BCCI Also BCC has, by definition, a

cubic lattice:a = (1, 0, 0)b = (0, 1, 0)c = (0, 0, 1)

I The atoms are located at:(0, 0, 0)( 1

2 , 12 , 1

2 )

I Each atom has 8 nearestneighbors and 6 next-nearestneighbors

I Packing density is 0.68I BCC metals are typically strong and reasonably ductileI Examples include: W, Cr, α-Fe and K

JELLIUM MODEL

I As the first approximation, a metal can be modeled as jelliumwhich consists of

1) a homogeneous electron gas, formed from the free electrons

2) a constant positive background density (ions)

I Hence, both electrons and ions are smeared to a respectiveeffective medium

I As a simple model of chemical bonding, placing (i.e.,“embedding”) a new atom into this medium, the change inenergy becomes

∆U = Uatom+jellium − (Uatom + Ujellium) ≡ ∆Uhom[n0] (1)

I Here, ∆Uhom[n0] is the embedding energy for the new atom intoa homogeneous electron gas with density n0

I Below is a list of density functional scheme examples from[Puska et al., PRB 24, 3037 (1981)]

I Obviously, for atoms with a closed electron shell (noble gases),the dependency is positive and linear→ repulsive interaction

I For other elements, there is a minimum in the curves whichleads to bonding in these materials

Effective Medium Theory

I In the effective medium theory (EMT) the interactions betweenthe particles are assumed to depend on an effective environment

I In this case, the total energy is written as

Utot =∑

iF [ni(Ri)] +

12∑

i,jΦ(Ri − Rj) (2)

I Here, F is a functional of the local electron density, and Φ is(typically) a purely repulsive pair potential for the atom cores

I In principle, EMT obtained from ab initio calculations can bedirectly used as a potential

I However, more typically, EMT is used as the starting point fordeveloping a more flexible potential

I Electron density n is approximated as a superposition of atom(or pseudo-atom) electron densities na(r)

n(r) =N∑

j=i+1na(ri − Rj) (3)

I The atomic densities na can be the densities for free atoms orfor atoms in a solid

I As a local approximation, the functional can be written as

∆Ui[ni] = ∆Uhom[ni(Ri)] (4)

I Total energy calculation for a complete system has beenpresented in [Manninen, PRB 34, 8486 (1986)]

I The total energy of the N-atom system is

UNtot = UN

R [n] (5)

I Here, R denotes that the functional depends on the sites Ri (andcharges Zi) of the nuclei (including the electrostaticnucleus-nucleus repulsion)

I This purely local model can be used to some extent, but it leadsto, e.g., wrong values for elastic constants [Daw, Baskes, PRB 29,

6443 (1984)]

I A better model is obtained by taking into account the electrondensity induced by an atom in the material

∆ρ(r) = ∆n(r) − Zδ(r) (6)

and by considering the difference between the real externalpotential and the jellium external potential δνext(r)

I This means that embedding an atom will change the localelectron density of the other atoms

I The first order correction to the homogeneous electron-gas term∆Ui[ni] = ∆U

hom[ni(Ri)] is

∆U(1)(Ri) =

∫d3r∆ρ(r − Ri)δν

ext(r) (7)

I This can be reduced to the electrostatic interaction between theinduced charge density of the embedded atom (∆ρ) and the totalcharge density of the system in which the atom is embedded:

∆U(1)(Ri) =

∫d3r∆ρ(r − Ri)φ(r) (8)

I φ is the total electrostatic potential of the system without theatom to be embedded

I The corrected total energy of the metal becomes

Utot =∑

iFhom[n̄(Ri)]+

12∑

i

∑j 6=i

∫d3r

∫d3r ′∆ρ(r − Ri)∆ρ(r ′ − Rj)

|r − r ′|(9)

I n̄ is an average density which takes into account the ∆ρcorrection

I The correction term is nowa sum over single-atompotentials ∆ρ(r − Ri)→we have obtained adensity-dependent termand a pair potential

I Comparison toexperiments shows adecent agreement:

Embedded-Atom Method (EAM)

I The EAM method [Daw, Foiles & Baskes, Mat. Sci. Rep. 9, 251 (1993)] isbased on the same basic ideas as the EMT

I Functional form is now deduced primarily semi-empirically andpartly by fitting

I EAM is physically less well-motivated than the EMT, but oftenyields better results

I The EAM total energy is written as

Utot =∑

iFi[ρi] +

12∑

ijVij(rij) (10)

I Here, ρi =∑

j 6=i ρaj (r) is the electron density at atom i; ρa

j (r) isthe density of atom j at distance r, and Fi is the embeddingfunction

I Clearly, this very much resembles the EMT total energy

I The embedding function F is universal, and the materialdependency comes in through the density ρ

I EAM is the most widely applied empirical potential for puremetals and alloys

I Daw, Baskes and Foiles obtained the functional forms for both Fand V by fitting to experimental results

I EAM pair potential is

Vij =1

4πε0

Zai (rij)Za

j (rij

rij(11)

where Zai is the screened charge for atom i

I Because of the simple pair-term, creating EAM potentials foralloys is straightforward

I Charge densities are obtained from ab initio calculations –typically a set of data points is given instead of an analyticfunction

I Basically, the EAM consists of three one-dimensional functionswhich are stored in an array and interpolated with splinefunctions

I For the original Cu potential, the functions are:

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1

1e+01

0 1 2 3 4 5

r (Å)

ρ(r) (eV/Å)

0.25

ρ (eV/Å)

-30

-25

-20

-15

-10

-5

0

0 0.05 0.1 0.15 0.2

F(ρ) (eV)

1e-04

1e-02

1

1e+02

1e+04

1e+06

1e+08

1e+10

0 1 2 3 4 5

r(Å)

U2(r) (eV)

Glue Models

I Glue models are similar to EAM, but with even less of thephysical interpretation

I For example, a gold potential by Ercolessi et al. [PRL 57, 719

(1986)], [Phil. Mag. A 58, 213 (1988)]

I The total potential energy is

U =12∑

ijφ(rij) +

∑i

U(ni) (12)

I φ is a two-body potential and U is the glue, which “providesgood energetics only for a properly coordinated atom”

I The density ni is considered to be an effective coordinationnumber (for Au @ T = 0 K: ni = n0 = 12)

I Functional forms for φ, ρ and U are obtained empirically, f.ex.

I This potential was originally developed to study surfacereconstructions in Au

I Note that the pair potential is no more purely repulsive

I The authors noted a good agreement with experimental results

Rosato Group Potentials

I In contrast to EAM and Glue Models which are based on DFT,the potentials originally introduced by Cleri and Rosato [PRB 48,

22 (1993)] have their origin in the tight binding (TB) method

I In TB, superpositions of the local atomic orbitals are used inelectronic structure calculations

I Motivation for using TB as the basis is that many properties oftransition metals can be derived from the density of states (DOS)of the outermost d electrons

I The first moment µ1 (average value) of the DOS fixes the energyand can be set to zero

I Square of the second momentõ2 gives the width of the DOS,

which is proportional to the binding energies in transition metals

I Electron d bands can be described by a basis of two-centerintegrals (“hopping integrals”; the matrix elements describe theoverlapping TB wave functions)

I The second moment of the electron DOS can be written as asum of squares of hopping integrals

µ2 = z(ddσ2 + 2ddπ2 + 2ddδ2) (13)

I The hopping integrals are functions of the inter-atomic distance,and the band energy for atom i can be written as

UiB = −

∑jξ2αβe−2qαβ(rij/rαβ

0 −1)

1/2

(14)

I This is formally similar to the embedding part of an EAM potentialif the square root operation is taken as the embedding function

I For stabilizing the crystal structure, a repulsive interaction is alsoneeded

I Typically, it is described by a sum of Born-Mayer ion-ionrepulsions

UiR =

∑j

Aαβe−pαβ(rij/rαβ0 −1) (15)

originating from the increase in kinetic energy of conductionelectrons constrained within two approaching ions

I Therefore, the parameter p must be related to the compressivityof the material

I Total energy becomes

Utot =∑

i(Ui

R + UiB) (16)

I Parameters A, ξ, p and q are obtained by fitting

I Despite the simple form, Rosato group potentials are applicablefor a wide range of FCC and HCP metals, and are known toproperly describe the elastic properties, defect energetics andmelting characteristics of the materials

I Later work [Mazzone et al., PRB 55, 837 (1997)] has extended themodel for alloys

I A word of warning: some of the Rosato group papers are knownto contain typos with minus signs and constant factors

Finnis-Sinclair Potentials

I One more group of EAM-like potentials are the so-calledFinnis-Sinclair potentials [Phil. Mag. A 50, 45 (1984)], [Phil. Mag. A 56, 15

(1987)]

I Also these transition metal potentials have their motivation in thetight binding method

I The functional form for the total energy is

Utot =12∑i 6=jφ(rij) − A

∑i

√ni, ni =

∑i 6=jρ(rij) (17)

I I.e., the same functional form as in the EAM withF [ni] = −A√ni

I The square root is motivated by the tight binding method, like inthe Rosato group potentials

I Parameters are obtained by fitting to experimental data

VACANCY FORMATION ENERGY REVISITED

I A simple estimate of vacancy formation energy:

Efvac = Etot(N, vac) − Etot(N, perfect). (18)

I EAM type potential

Utot =∑

iFi[ρi] +

12∑

ijVij(rij) (19)

I Here, ρi =∑

j 6=i ρaj (r) is the electron density at atom i

I At a perfect fcc lattice, with only NN interaction, equilibrium bondlength:

Etot(N, perfect) = NF [12ρ0] +1212Nφ = NF [12ρ0] + 6Nφ

(20)

Etot(N, vac) = (N−12)F [12ρ0]+12F [11ρ0]+12 [(N−12)12φ+12×11φ]

(21)

I where ρ0 = ρa(r0),φ = V(r0)

→ Efvac = 12[F(11ρ0) − F(12ρ0)] − 6φ (22)

I Cohesion energy per atom is

Ecoh =Etot(N, perfect

N = F(12ρ0)+6φ→ 6φ = Ecoh−F(12ρ0)

(23)

I Substituting 6φ into Efvac :

Efvac = 12F(11ρ0) − 11F(12ρ0) − Ecoh (24)

I Note: for a pair potential bonding scales linearly with the numberof bonds:

Efvac = 12(11U0) − 11(12U0) − Ecoh = −Ecoh (25)

I So with EAM we at least have Efvac 6= −Ecoh

Examples of EAM Potentials for FCC

I Original EAM potentials for FCC metals [Foiles, PRB 32, 3409 (1985);ibid 33, 7983 (1986)]

I Good potentials for Ni, Cu, Pd, Ag, Pt, Au and alloys of these

I Potentials yield reasonable properties except for surfaces (atypical problem for EAM type potentials due to averaged density)

I Decent potential for Al

I Much used and widely tested, no other obvious shortcomings

Examples of EAM-like Potentials for FCC

I Glue potentials: [Ercolessi & Adams, Europhys. Lett. 26, 583 (1994)]: goodpotentials for Au and Al

I Alternative EAM potentials for Al, Ag, Au, Cu, Ni, Pd and Pt: [Cai

& Ye, PRB 54, 8398 (1996)] – possibly better description of alloys andsurfaces than with the original one

I Analytical EAM for most FCC metals: [Johnson, PRB 37, 3924 (1988)]

I Potential for Cu, Ti and their alloys: [Sabochick & Lam, PRB 43, 5243

(1991)]

I A Cu potential with very good point defect properties: [Nordlund &

Averback, PRL 80, 4201 (1998)]

I NiAl-systen potential: [Voter & Chen, Mat. Res. Soc. Symp. Proc. 82, 175

(1989)]

I Cleri-Rosato parameters for FCC metals exist for Ni, Cu, Rh, Pd,Ag, Ir, Pt, Au, Al and Pb (see above)

I EAM potentials for Cu, Ag, Au and alloys: [Ackland & Vitek, PRB 41,

10324 (1990)], improvements: [Deng & Bacon, PRB 48, 10022 (1993)]

Examples of EAM-like Potentials for HCP

I Relatively little work done on HCP as compared to FCC/BCCmetals - stabilized by a more complicated pair potential

I The difficulty is that HCP must be lower in energy than FCC, andthat the c/a ratio should be about right

I Also, there are 5 elastic constants instead of the three for cubiccrystals

I HCP potential for Hf, Ti, Mg and Co: [PRB 45, 12704 (1992)],however, they also showed that for metals which have

c13 − c44 < 0 or (1/2)(3c12 − c11) < c13 − c44 (26)

the EAM-type potentials won’t work (e.g., Be, Y, Zr, Cd and Zn)I Cleri & Rosato derived parameters also for HCP Ti, Zr, Co, Cd,

Zn and MgI Also Johnson & Oh developed HCP potentials (Mg, Ti, Zr):

[J. Mater. Res. 3, 471 (1988)]

Examples of EAM-like Potentials for BCC

I In BCC potentials, the second-nearest neighbors must be takeninto account

I For BCC, Finnis-Sinclair–type potentials have been widely usedI Originally, e.g., Fe, V, Nb, Ta, Mo and W were developed. Fatal

errors with some of these were later corrected: [Ackland & Thetford,

Phil. Mag. A 56, 15 (1987)]

I Li, Na, K, V, Nb, Ta, Cr, Mo, W and Fe potentials by Johnson &Oh: [J. Mater. Res. 4, 1195 (1989)]

I One study has proposed that 4-body terms could be crucial forBCC materials (not included in EAM): [Moriarty, PRB 42, 1609 (1990)]

I 4-body potential has been developed for at least a few metals,see: [PRB 49, 124310 (1994)], [PRB 54, 6941 (1996)]

Metal-hydrogen–hybrid Potential

I It’s also possible to develop a EAM-like potential for hydrogen inmetals

I They can reasonably produce, e.g., cohesive and migrationenergies of hydrogen in solid metals and hydrogen on solidsurfaces

I One example is presented for Ni-H in: [Rice et al., J. Chem. Phys. 92,

775 (1990)]

EAM and Repulsive Potentials

I As has been said during the previous lectures, for high energyevents (K > 10 eV), the repulsive potential has to be almostalways corrected at short interatomic distances

I One option is to use a universal repulsive ZBL potential (alsoalready mentioned), another one is to utilize all-electron ab initiodata

I For EAM-type potentials the electron density needs to be fixed toa constant value where the repulsive potential is used

I Example: Foiles Pd-potential has been joined to the ZBL; thehigh-pressure properties and melting point of the potential wereobtained almost exactly right, interstitial atom energetics becameworse [Nordlund et al., PRB 57, 13965 (1998)]

Two-band EAM

I EAM is often used for transition metals where the bondingoccurs via s and d orbitals (bands)

I However, in the standard form of EAM, the occupations of thesebands are included in the density function in an arbitrary manner

I It is also possible take this into account explicitlyI In this case, the energy of atom i is written as

Ui = Fd(ρi,d) + Fs(ρi,s) +12∑

jV(rij) (27)

where the subscripts stand for the electron density contributionsof each band

I A two-band model has been developed for caesium [PRB 67,

174108 (2003)] and the binary alloy FeCr [PRB 72, 214119 (2005)]

I For many transition metals the cohesion is determined mainly bythe d band, but the s band affects the elastic properties viarepulsion

I On the other hand, alkali and alkaline-earth metals are normallyclose-packed with bonding determined by the s electrons

I However, at large pressures the electrons are transferred to themore compact d bands although they are higher in energy

I With two-band models, e.g. isostructural transition of Cs andthermodynamical properties of FeCr alloy were reasonablydescribed

FINAL NOTE ON METALS

MD does not really have electrons!

I No electronic heat capacity

I No electronic heat conductivity

I No energy exchange between electronic and ionic subsystems:electronic stopping or electron-phonon coupling

If needed, these need to be taken into account by artificial methods

SUMMARY

I Metals are characterized by the free electrons which bind to thepositive nuclei

I Therefore, local density approximations in the spirit of densityfunctional theory are well suitable for describing them

I From the physics point-of-view, the so-called effective mediumtheory (EMT) gives a reasonable picture of metals

I However, by using emperically obtained functionals, a betteragreement is obtained with experiments

I The most successful approach using this approach is theembedded-atom method (EAM)

I It has been applied to a wide variety of metals both in its originalform and with several modifications