Temperature dependence of the range of focused collision sequences in copper single crystals

Abstract

The well-known focusing model, which is responsible for the [110] spots, observed in the sputtering of a fcc monocrystal describes a series of collisions along a closed packed atom row in fcc metals. The range of such a focusing sequence has been calculated by Leibfried for copper, using a Born-Mayer potential. For the colliding atoms a hard-sphere model is used. This range is determined from the energy lost at every collision in the sequence by potential interaction with the neighbouring [110] rows. In this article we have generalized Leibfried's treatment by considering the influence of the thermal vibrations of the crystal atoms on the energy loss per collision above the Debye temperature. In doing so, we have made the approximation that the thermal vibrations are harmonic and the coupling between nearest and next-nearest neighbours only has been taken into account. For the coupling constants we have taken values, derived by Jacobsen from experimental results. Using these values in a machine calculation, we compute the mean square of the difference of displacements of two nearest neighbours as a function of temperature. The relative energy loss factor can now be calculated as an average over these displacement differences. The result is ε( T) = ε(0). exp(23.10 −4 T), where ε(0) is the relative energy loss factor as calculated by Leibfried for a non-vibrating lattice. In the case of gold Nelson, Thompson and Montgomery have calculated the relative energy loss per collision due to a different mechanism, namely the scattering of momentum out of the sequence, because of the fact that the thermal vibrations cause the centres of the colliding atoms to be not exactly in one line. We have repeated their calculation for copper and compared the result with the one obtained above for the potential interaction with the neighbouring [110] rows. By treating the two effects as additive we obtain the range of focusing sequences as a function of temperature and initial energy. It turns out that these ranges are much smaller than was previously expected. We find a maximum range of 12 at 315°K and of 4 at 900°K (cf. fig. 5). On the basis of these results we can give a qualitative explanation of some features of the sputtering of single crystals. Because of the shortness of the average focusing range for sputtering which is 5 at 315°K and 2 at 900°K there will be a significant competition between focusing and defocusing mechanisms in ejecting the sputtered particles. From this we conclude that the amount of material, deposited in the [110] focused spots will have a weaker dependence on the range of the focusing sequences than a direct proportionality. We can because of this, expect the total sputtering ratio to vary only slightly with temperature above the Debye temperature.