Question

Visualize a gold thin film stripe as a connector in a microelectronic chip. Suppose the strip is 0.1 µm thick and has a bamboo grain structure. (This means the grain boundaries run laterally completely across the stripe.) Take the grain boundary energy of gold at 600 K to be 420 (erg/cm2 ); its surface energy is 1440 (erg/cm2 )

a. Compute the dihedral angle where a grain boundary meets the external surface.

b. Find a critical grain boundary spacing, sc, for which the equilibrium grain shape will produce a hole in the film. (Answer to b is  4.1µm. Please supply the correct proof)

Answer 1

(a) Compute the dihedral angle where a grain boundary meets the external surface.

using a formula, we have

( _{$\gamma$ Gb} / _{$\gamma$ S}) = 2 cos ( _$\Psi$ / 2)

where, _{$\gamma$ Gb} = grain boundary energy of gold = 420 erg/cm²

_{$\gamma$ S} = surface energy = 1440 erg/cm²

then, we get

[(420 erg/cm²) / (1440 erg/cm²)] = 2 cos ( _$\Psi$ / 2)

[(0.2916) / 2] = cos ( _$\Psi$ / 2)

(0.1458) = cos ( _$\Psi$ / 2)

cos^-1 (0.1458) = ( _$\Psi$ / 2)

(81.6 degree) = ( _$\Psi$ / 2)

_$\Psi$ = 163.2 degree