Response to 'Comment on 'Effect of current crowding on vacancy diffusion and void formation in electromigration'' Appl. Phys. Lett. 79, 1061 (2001)

摘要

Lloyd made three comments. The first is on Z~(*) of vacancies. The second is on the gradient force in the ac mode of electromigration. And the third is on the effect of steady state stress on vacancy diffusion and void formation. First, about Z~(*) of vacancies, if we assume in a facecentered-cubic metal that the atomic lattice diffusion is mediated by a vacancy mechanism, the diffusion of an atom to the right-hand side is the same as a vacancy to the left-hand side. The configuration of the activated state of a diffusing atom is the same whether you consider the atom moving in one direction or the vacancy moving in the opposite direction. On the basis of Huntington's model of electromigration, Z~(*) is basically the ratio of the specific resistance of a diffusing atom and that of a normal lattice atom. Huntington used the idea of momentum exchange (electron wind force) to derive an expression of Z~(*). In other words, a defect (a vacancy or an impurity atom) which causes a larger resistance in the current flow will experience a larger electron wind force to push it away; it is an argument of reaction and counter reaction. The momentum exchange depends on the cross section of scattering. If we take a simplified picture and look at the activated state of a diffusing atom (or vacancy), its cross section of scattering is about a factor of ten greater than that of a lattice atom. Now, if we multiply that by the valence of Al, which is three, we have an effective charge number of about 30 for an Al atom during electromigration. For a monovalence metal such as Cu, the Z~(*) is about 10, and for a quadrivalence metal such as Pb, the Z~(*) is about 40. So the physical meaning of Z~(*) is only for calculating the magnitude of the driving force in electromigration; it does not mean that the diffusing atom or vacancy has a charge of Z~(*) e.