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Effect on Elasticity

There are several ways of calculating the elastic properties of a material from the knowledge of the interaction potential. They can be found either from gathering statistics from long molecular dynamics calculations using the fluctuation formula method[107], or directly from inverting the matrix of spring constants which can easily be calculated from the interaction potential. The advantage of using the molecular dynamics method is that the elastic constants can be calculated at temperature and the other thermodynamic averages can be calculated simultaneously. Obviously, inverting a tex2html_wrap_inline7044 matrix at every timestep in the simulation of temperature is computationally prohibitive, but the calculations presented here are carried out at 0K, therefore making the latter method more effective since the dynamical matrix has already been calculated.

A full derivation of the calculation of elastic coefficients can be found in reference [114]. The final method for a crystal containing many atoms in a single primitive unit cell will be given here. The elastic coefficients can be calculated directly from tex2html_wrap_inline7168 as follows. Allow tex2html_wrap_inline5398 , tex2html_wrap_inline5370 , tex2html_wrap_inline7174 , tex2html_wrap_inline7176 , tex2html_wrap_inline5754 and tex2html_wrap_inline5496 to run over the co-ordinate axes x, y and z. Then define

    eqnarray2163

There will be three modes that have zero frequency due to the three translational degrees of freedom of the entire crystal. Therefore tex2html_wrap_inline7188 will be singular since it will have linearly dependant rows and columns. Now we introduce the tex2html_wrap_inline7190 matrix tex2html_wrap_inline7192 which is defined to be the inverse of tex2html_wrap_inline7188 , where we allow tex2html_wrap_inline7196 , and for convenience of calculation define

eqnarray2180

For convenience, the k=1 rows and columns are removed from the dynamical matrix for the inversion. The choice of k is arbitrary and therefore can be taken to be k=1 without loss of generality.

Now define the following brackets where tex2html_wrap_inline7204 is the volume of the supercell:

  eqnarray2187

and

  eqnarray2194

The elastic coefficients are then

eqnarray2206

provided the strain energy of the crystal is invariant against rigid body rotations, the force on each atom is zero and the stresses on the crystal vanish. The atomistic relaxation simulation which relaxed the atoms into the lowest local energy configuration and the periodic boundary conditions on the supercells in conjunction with the Parrinello-Rahman Lagrangian ensures that these conditions are satisfied.

The elastic coefficients can then be expressed in their more familiar form by pairing the indices by the equivalences tex2html_wrap_inline7206 , tex2html_wrap_inline7208 , tex2html_wrap_inline7210 , tex2html_wrap_inline7212 , tex2html_wrap_inline7214 and tex2html_wrap_inline7216 .

The bulk modulus, tex2html_wrap_inline7218 (for a cubic crystal) is then

eqnarray2210

Table gif shows the independent elastic coefficients for the various supercells in the two simulations. The entire tex2html_wrap_inline7220 elastic matrix is calculated in all cases. It is found that tex2html_wrap_inline7222 , tex2html_wrap_inline7224 and tex2html_wrap_inline7226 . All other coefficients are almost zerogif. The bulk modulus is higher than that of experiment ( tex2html_wrap_inline7218 =0.61 eV/Å tex2html_wrap_inline5346 ) because the parameters of the potential were fitted for C=0gif, although the elastic coefficients are sensitive to the value of C. It can be seen, however, that the change in bulk modulus seems to be almost independent of the type of defect under consideration. The elastic coefficients are also calculated for the smaller supercells, which allows the change in elastic coefficients with respect to oncentration to be calculated. The coefficients are seen to rise with concentration, (Figure 7.13),apart from C tex2html_wrap_inline5360 which decreases very slightly. This shows the characteristic defect stiffening in silicon[74, 75].

  figure2235
Figure 7.13: Change in elastic coefficients with respect to concentration are shown for the hexagonal interstitial. It is found that the change in elastic coefficients with respect to concentration, tex2html_wrap_inline5358 , is fairly constant over the range considered. Note the change in scale for C tex2html_wrap_inline5360 . 


next up previous
Next: Conclusions Up: Empirical Calculations Previous: Free Energy and Entropy

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996