A modified embedded atom method study of the high ...

Materials Chemistry and Physics 80 (2003) 405–408

Material science communication

A modified embedded atom method study of the high pressure phases of silicon N. Badis a,∗ , H. Feraoun a , H. Aourag a,b , M. Certier c b

a Computational Materials Science Laboratory, Université de Sidi-Belabbes, Algeria LERMPS, Université de Belfort-Montbeliard, Site des Sevennans, 90010 Belfort, France c LPLI, Université de Metz, France

Accepted 15 March 2002

Abstract We report the results of a modified embedded atom method (MEAM) study of the high pressure phase diagram of silicon especially for the bc8, face-centred cubic (fcc), body-centred cubic (bcc) and simple cubic (sc) phases. Our results are in good agreement with the available experimental data and the first principles theoretical studies. © 2002 Published by Elsevier Science B.V. Keywords: Silicon; Modified embedded atom method; High pressure

1. Introduction Theoretical studies of cohesive, structural and vibrational properties of semiconductors under pressure are now routinely being performed by means of ab initio calculations. The accuracy of total energies obtained within the density functional theory, often even using the local approximation, is in many cases sufficient to predict which structure, at a given pressure, has the lowest free energy. Semiconductors exhibit a multitude of structural transformations when pressure is applied, and the study of the various high pressure phases has become a central field in theoretical and experimental solid state physics. While the diamond structure of silicon is the most widely studied of all semiconductors, there have also been a number of experimental and theoretical studies of the high pressure phases of silicon [1–8]. A detailed picture of the phase diagram of silicon has emerged, although a number of unanswered questions remain. Experimental studies have used the diamond-anvil technique [9], which can achieve pressures of at least 2500 kbar, together with energy dispersive X-ray diffraction techniques which gives information about the structures formed. A large number of theoretical studies have been performed using the first principles pseudopotential total energy methods [4–8,10,11], although other studies have also been reported using a self-consistent linear com∗ Corresponding author. Present address: LERMPS, Universit´ e de Belfort-Montbeliard, Site des Sevennans, 90010 Belfort, France.

0254-0584/02/$ – see front matter © 2002 Published by Elsevier Science B.V. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 1 3 3 - 5

bination of atomic orbitals (LCAOs) technique [12] and the linearized muffin tin orbital (LMTO) method [13]. The homopolar (zero ionicity) semiconductors undergo very many transformations under pressure. On decrease of pressure from ␤-Sn–Si a strongly distorted tetrahedral phase with eight atoms in a rhombohedral cell is first obtained (R8). On further decrease of pressure, this phase transforms to a body-centred cubic (bcc), structure with eight atoms in the bcc cell (bc8) which persists during long periods of time at ambient conditions. In spite of greatly increased computer speeds, the application of ab initio methods for an atomistic simulation of materials is still limited to relatively small ensembles of atoms and, in molecular dynamics, relatively short simulation times. In contrast, the use of empirical or semiempirical interatomic potentials makes it possible to simulate much larger systems for much longer times, and thus tackle such problems as plastic deformation, fracture, or atomic diffusion. For this reason there is and will probably always be a demand for realistic interatomic potentials as there will always be a tendency to simulate systems as large as possible. The modified embedded atom method (MEAM) potential proposed by Baskes and coworkers [14–16] is be said to be unique in that it can reproduce properties of many metals and semiconductors with various crystal structures including hexagonal close packed (hcp) and diamond cubic (cd) as well as face-centred cubic (fcc) and bcc using the same formalism. It is therefore interesting to verify the transferability of this potential to the high pressure phases of silicon.

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N. Badis et al. / Materials Chemistry and Physics 80 (2003) 405–408

Table 1 The parameters used in the construction of the MEAM potential Ec

Re

α

A

β(0)

β(1)

β(2)

β(3)

t(1)

t(2)

t(3)

ρ0

4.63

235

4.87

1.00

4.40

5.5

5.5

5.5

3.13

4.47

−1.80

2.05

The details of the method are given elsewhere [14]. The adjusted data used for the parametrization of the potential are given in Table 1.

2. Calculations The embedded atom method is a technique for the construction of many-body potential models for metals by combining the density functional formalism with traditional pair potentials. The key idea is to divide the internal energy into a density dependent term that accounts for the cohesion due to the electron glue in which ions are immersed, and a pairwise term that should contain the core repulsion. Accordingly, the basic equations of the method are. Etot =

N


Fi (ρh,i ) +

i=1

ρh,i =

N


N 1
Φij (rij ) 2

(2)

j(j=i)

Fi is the embedding function of atom i in the background electron density ρ¯ i and is given as a simple function of this one: F(ρ) ¯ = AEc ρ¯ Ln ρ¯

j=1(j=i)

where Etot is the total energy of the system of N atoms, Fi (ρh,i ) the embedding energy required to place an atom i in the electron density ρh,i due to all neighboring atoms (host electron density), rj the contribution of j atom to the electron density as a function of the distance to its centre and Φij is the pairwise interaction between atoms i and j as a function of their separation rij . Eq. (2) suggests that the host electron density is formed by a linear superposition of atomic contributions, this assumption is justified by arguments from the effective medium theory. The embedding energy is a universal functional independent of the origin or the distribution of the electron density. In the case of monoatomic metals, all atoms of the structure are identical so that the total energy of the system is Etot = NE

(3)

where E is the energy per atom and N is the total number of atoms in the system. So the atomic contribution to the host electron density, the embedding energy and the pairwise interaction would be identical, one can then write   N N
1
E = F  ρi (rj ) + Φij (rij ) (4) 2 j=1

i

(1)

j=1(j=i)

ρj (rij )

[14], the extension is “empirical and has not been justified by strong physical arguments as have the EAM and the N-body potential method”. Actually, what motivated the modification of the EAM scheme was the search of a model fitting to both metallic and covalent environments that would offer a large opportunity to describe systems where dissimilar materials are on hand. Hence, it is especially in the study of semiconducting materials and metal–semiconductor interfaces where the MEAM was intensively used. The MEAM follows the EAM concept in that the energy of an atom is taken as one half the energy in two body bonds with its neighbors plus the energy to embed the atom in the electron density at its site arising from all other atoms:  

1 Fi (ρ¯ i ) + Φij (rij ) (5) E= 2

where Φij (rij ) is the pair interaction between atoms i and j at a distance rij . However, while in the EAM the electron density is a simple sum of radially dependent contributions from the other atoms, in the MEAM it includes angular dependence: it contains the spherically symmetric partial elec(0) tron density ρi as in the EAM, plus angular contributions (1) (2) (3) ρi , ρi and ρi :  2
(0) a(0) (7) ρj (rij ) (ρi )2 =  j(j=i)

(1)

(ρi )2 =

(2) (ρi )2

=




 

α














α,β

 

a(1)

ρj

β

rijα rij

j(j=i)



2



rij

j(j=i)



rijα

(rij )



(8)

2

 ρa(2) (rij ) j rij2

2 1 
a(2) − ρj (rij ) 3

j=1(j=i)

where rj is the distance between the atom and its jth neighbor, the sum being extended to all those neighbors. The MEAM is an empirical extension of the EAM that includes angular forces. As stated by the method’s author in

(6)

(9)

j(j=i)

(3)

(ρi )2 =


α,β,γ

 


j(j=i)

 

β γ

rijα rij rij rij3



2

 ρa(3) (rij ) j

(10)

N. Badis et al. / Materials Chemistry and Physics 80 (2003) 405–408 a(l)

Here, ρj is the contribution of j atom to the electron density at a distance rij from site i, given by a(l)

ρi (r) = ρe e−β

(l) (r/r −1) e

constants and bulk modulus as well as the fcc − bcc energy difference.

(11)

and rijα is the α component of the distance vector between atoms i and j. These partial contributions can be combined to give the total background electron density using several ways. One physically meaningful form is given by 

(l) 
1 (0) (l) ρi  ρ¯ i = ρi 1 + (12) ti (0) 2 ρ i l=1,3 (l)

407

where ti are the weighting factors relative to each partial density contribution. This representation exemplifies that the additional angular terms are perturbative adjustments to the spherical density due to the existence of gradients. Our choice toward the MEAM was twofold. First, the determination of the pair interaction (see below) involves principally the reference structure and the universal equation of state so that there is no need for further parametrization that would introduce some arbitrariness in the potential definition. Secondly, the directional bonding involved in the MEAM expression permits a better representation of the inhomogeneous environment around an atom when its surrounding neighbors are from different species. The use of the MEAM entails the knowledge of 12 parameters, namely E0 , re and α involved in the universal equation of state and directly taken from experiment as explained in the previous section, the factors of the exponential decay of the atomic densities β(l ) (l = 0, 1, 2, 3), the partial angular density weights t(l ) (l = 0, 1, 2, 3) and the remaining parameter A introduced in the expression of the embedding energy. Those parameters can be determined given the reference lattice structure, cohesive energy, atomic volume, shear

3. Results and discussion The structural phases considered in this paper are cd, fcc, bcc, simple cubic (sc) and the bc8. Details of most of these structures can be found in [17] and in the references therein. Graph of Murnaghan equation fits to energy against volume for silicon is shown in Fig. 1. This graph shows that the relaxed structures are unstable with respect to diamond at low pressures. For all phases, the calculated results for the equilibrium structural parameters are in fairly good agreement with those reported in previous studies. The lattice parameters and bulk modulus as determined fits to a Murnaghan equation of state:

 B  BV 1 V0 E=  + 1 + E0 B B − 1 V where B = −∂2 E/V∂V 2 is the bulk modulus and B is its pressure derivative. These are given in Table 2. A rather small difference is observed between our calculated and those of previous calculations, especially for the bc8 phase. This may be due in one hand to the use of the Murnaghan equation of state. Since there is a debate as to whether a Murnaghan fit is appropriate in these structures, some of the calculated points are at unphysical compression of up to 30%. The approximation that dB/dP is a constant throughout this huge pressure range is extremely doubtful. On the other hand, this difference may be also due to the fact that the MEAM gives some crude results concerning the bcc structures, and the extension of the method to the second nearest neighbors should be necessary [18].

Fig. 1. Energy–volume curves for different phases of Si.

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N. Badis et al. / Materials Chemistry and Physics 80 (2003) 405–408

Table 2 The structural and elastic properties or silicon at different high pressure phases a0 (Å) Diamond bc8 fcc bcc sc

(c11 −c12 )/2 (Mbar)

B0 (Mbar) 5.43a ,

5.429b

5.431, 5.065, 6.752d 4.15, 4.19e , 3.89f 3.19, 3.25e , 3.09f 2.65, 2.61e , 2.53f

0.976c

0.977, 0.66, 0.589d 1.04, 1.64e 0.104, 0.11e 0.84, 0.89e

0.503,

0.504b

0.41, 0.35e 0.133 1.65, 2.15e

c44 (Mbar) 0.776, 0.789b 0.891, 0.91e 0.182 0.16, 0.18e

a

From Ref. [20]. From Ref. [21]. c From Ref. [22]. d From Ref. [23]. e From Ref. [24]. f From Ref. [25]. b

The elastic constant results (see Table 2) show also a good agreement with those of previous studies. The structures are elastically stable since the stability condition c44 > 0, c11 > 0 and c11 > c12 are satisfied. 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. [19]. Therefore, we may conclude that a simple MEAM potential can reproduce with a fairly good approximation the high pressure phases of silicon. References [1] [2] [3] [4] [5]

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