5. covalent bonding

8/9/2019 5. Covalent Bonding

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POTENTIALS FOR COVALENT SOLIDS -SEMICONDUCTORS

The covalent bonding between atoms arises naturally in complete quant

mechanical treatments of the cohesion, such as calculations based on the denfunctional theory (DFT) where the covalent bonding is treated practically exacHowever, as in many other cases, use of these methods in studies of complex structof defects may not be feasible.

Hierarchically, the next possible step is the tight-binding approximation whaccounts for the electronic structure of the system but without self-consistent evaluaof the relevant Hamiltonian. Rather, the Hamiltonian is fixed and parameters in itfitted empirically. The first such study was made by Chadi 1. This approach capturesthe following quantum mechanical features of bonding:

The energy of a system does not depend only on the separation of atoms but alsoon the angles between chemical type bonds formed between pairs of atoms.Hence, forces acting on atoms also depend on angles between bonds.

However, even tight-binding calculations are frequently very extensive and tconsuming. For this reason, empirical potentials of bonding in covalent solids hbeen developed that incorporate the most important features of the directional charaof bonding in these materials. We review here briefly Stillinger-Weber potential

Tersoffs potential, originally developed for silicon but generalized to other covamaterials2. The most important characteristic that has to be reproduced by all supotentials is preference for tetrahedral configurations of atoms and related diamcrystal structure.

1 D. J. Chadi, 1979, Phys. Rev. B19, 2074; 1984, Phys. Rev. B29, 785 2 F. Stillinger and T. Weber, 1985, Phys. Rev. B31, 5262; J. Tersoff, 1988, Phys. Rev. B37 ,6991;38 , 9902

Diamond type structure of silicon

Covalent bonds are marked byconnecting lines

Dark atoms and bonds show thetetrahedron with 109.7o angles of


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Stillinger-Weber potential

The potential energy of a system of N interacting particles can be generally wrias

E p = V2 (i, j) +i, j! V3 (i, j, k) +

i, j,k ! V4 ( i, j, k, ! ) + ....

i, j,k, !! (S1)

where V2 describes the two-body (pair) interactions, V3 three-body interactions, V4 four-body interactions etc. In the scheme proposed by Stillinger and Weber only tand three-body interactions are included.

The two-body interactions are described by a pair potential such that

V2(i, j)= ! "

2 (r ij /#

) (S2.1)where the functional form

!2 (r ) = A(Br

" p " r " q )exp[(r " a ) " 1 ] for r < a

!2 (r ) = 0 for r # a

(S2.2)

! is set such that the depth of" 2 is 1 and# is chosen such thatd! 2 (r = 21/6 ) dr = 0 .

A, B, p, q anda are all positive adjustable parameters (Note:a is not a latticeparameter). This form automatically cuts-off the potential and all its derivatives ata ; note that for r


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The seven parameters A, B, p, q,a , $ and % are all fitted to assure that the diamondstructure is the lowest energy structure and the melting temperature is in a reasonagreement with experiment. For silicon the most satisfactory parameters were founbe:A = 7.049556277, B = 0.6022245584, p = 4, q = 0,a = 1.80, $ = 21.0 % = 1.20Figure 1a shows the corresponding binding energy per atom of Si (in reduced unversus number density (in reduced units) for several simple structures: f.c.c, b.simple cubic (s.c.) and diamond. In order to obtain correct lattice spacing and coheenergy of the diamond structure the parameters# and ! have to be chosen as# =209.51nm ! = -50 kcal/mol. Figure 1b shows the corresponding binding energy patom of Si calculated using a DFT based method. (Note that in Fig. 1a the x axdensity and in Fig. 1b volume and these two quantities are related inversely).

The Stillinger-Weber potential does assure that the diamond structure is the lowenergy structure but agreement with the DFT calculations is only tentative but the oof energies for different structures is reasonable.

(a) (b)

Fig. 1. (a) Energy versus density dependencies for diamond, fcc, bcc and simple c(sc) structures of silicon calculated using the Stillinger-Weber potential. (b) Eneversus volume dependencies for diamond, fcc, bcc and simple cubic (sc) structuresilicon calculated using a DFT method3.

3 M. T. Yin and M. L. Cohen, 1982,Phys. Rev. B 26 , 5668; 1984,Phys. Rev. B 29 , 6996


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Tersoff potential

The most appropriate empirical schemes for covalently bonded solids are thwhich reflect correctly the angular dependence of atomic interactions that isconsequence of the quantum mechanical nature of bonding. One of the most succe

empirical schemes of this type is Tersoff's potential, originally developed for siliconis based on the concept ofbond-order that is a measure of the variation of the strengthof a bond between two atoms with the surrounding atomic environment, in particuthe coordination number. Within this scheme the total energy is decomposed into parts

E p = U rep + U bond (S4)

The first term is the short-range Pauli repulsion and the second term the bonding teThe repulsion is represented by a pair potential,! , so that

U rep =12 !

i" j# (rij ) (S5)where rij is the separation between atoms i and j. In the Tersoff's scheme the ppotential is written as

! (r ij ) = A f cut (rij ) exp( "# 1 rij ) (S6.1)where

f cut (rij ) =

1, rij < R ! D12 ! 12 sin

"2 (rij ! R ) D[ ]

0, rij > R + D

#

$ % %

& % %

, R ! D < rij < R + D (S6.2)

is a cut-off function that limits the range of the interactions to the first nearest neighshells in the diamond lattice and A,! 1 , R and D are constants determined within theempirical fitting.

The bonding term is proportional to the bond-orders, b ij , relating to the covalent bondsformed between atoms i and j, and is written as

U bond = =12 b ij FA (r ij )

i! j" (S7.1)

whereFA (r ij ) = ! B f cut ( rij ) exp( !" 2 rij ) (S7.2)

is a rapidly decreasing function of the separationsrij and B and! 2 are again adjustableparameters. The bond-order,b ij , represents the many-body, non-central character oatomic interactions and is taken as a monotonically decreasing function of


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coordinations of atoms i and j. In the latest version of the Tersoff's potentialsb ij hasthe following form

b ij = 1 + pn ! ij

n( )" 1 2n

(S8.1)where

! ij=

f cutk " i,j# (r ik ) exp[ $ 33

(rij % rik )3

] g(&ijk ) (S8.2)

The summation extends over all the neighbors k of the atoms i and j; p, n and! 3 areadjustable parameters andg(! ijk ) is a function of the angle! ijk between the i-j and i-kbonds. This function has been parameterized as

g( ! ijk ) = 1 + c2 d 2 " c2 d 2 + (h " cos ! ijk )

2[ ] (S8.3)where c, d, and h are empirically fitted constants.

The parameters figuring in the Tersoffs potential were determined by fitting cohesive energy, lattice spacing and bulk modulus of silicon in the diamond latticethe volume dependence of the energy of several other structures (hcp, bcc,&-tin etc.)predicted from first-principles pseudopotential calculations3. These dependencies areshown in Fig. 2. All the fitting parameters involved in the Tersoff's potential for silare summarized in the table shown below.

A (eV) 1.8308 x10 3 c 1.0039 x10 5 B (eV) 4.7118 x10 2 d 1.6218 x10 1

! 1 () 2.4799 h -5.9826 x10! 1

! 2 (-1) 1.7322 ! 3 (-1) 1.7322p 1.0999 x10 ! 6 R () 2.85n 7.8734 x10 ! 1 D () 0.15

An analogous potential was developed by Brenner4

for carbon and hydrocarbons.It was used, for example in molecular dynamics calculations of chemical vadeposition of diamond films.

4 D. W. Brenner, 1990,Phys. Rev. B 242 , 9458.


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(a)

(b)Fig. 2 Comparison of energy versus volume dependencies for diamond, fcc, bcc simple cubic (sc), simple hexagonal (sh) and&-tin structures of silicon calculated using:(a) DFT based pseudopotential method3; (b) Tersoff potential.


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In electronic structure calculations based on the density functional theory (DFT)interesting finding is that at the compression at which the volume per atom decreaseabout 16-17%, i.e. when the volume per atom becomes about 16.53, the diamondstructure starts to have a higher energy than the structure corresponding to& tin (see

Fig. 2a). Hence calculations predict that at this compression there is a phtransformation from the diamond to the& tin structure. This transformation was,indeed, observed experimentally. The interesting feature of the Tersoff potential is it reproduces this aspect of silicon quite well, as seen from Fig. 2b.

5. covalent bonding

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