The Influence of Strain Energy Minimization on Abnormal Grain Growth in Copper Thin Films
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dr dt
=
gby-1 + I
Tr
,
-+ h
h
(1)
1
where r is the radius of the growing grain, M is the average grain boundary mobility, ?gb is the average grain boundary energy, F is the average grain radius, h is the film thickness and T., Ti, 7,y and y1 are the average surface and interface energies of the matrix of surrounding grains and of the growing grain, respectively. The first term in the sum is the classic driving force for normal grain growth and results from reduction of energy by elimination of grain boundaries. The other two terms are unique to the thin film geometry and are due to minimization of the energy of the surface and film-substrate interfaces. Since interface energy is typically highly anisotropic, these driving forces determine the orientation of the growing grain. For fcc metals on amorphous underlayers, this theory predicts a (11l) texture for abnormally growing grains, since these are the most densely packed planes. 103 Mat. Res. Soc. Symp. Proc. Vol. 391 ©1995 Materials Research Society
The above theory does not predict experimental observations of abnormal grain growth of certain orientations, namely the (110) and (112) in Al alloy films and the (100) in Cu films. To explain the (110) texture in Al films, Sanchez and Artz [20] proposed that a difference between the average strain energy density of the matrix of grains and the growing grain produces a driving force that is sensitive to the orientation of the growing grain. Although this theory originally focused on strain energy differences induced by orientationally dependent plasticity, Thompson [21] noted that such a driving force could also be important in elastically anisotropic materials, such as Cu. The development of a biaxial strain upon heating a thin film attached to a substrate with a dissimilar coefficient of thermal expansion is well known [22]. This applied thermal strain is given simply by AaAT, where Aa is the difference in thermal expansion coefficients between the film and the substrate and AT is the difference between the annealing temperature and the zero stress temperature. For Cu, where the biaxial modulus for the (111) orientation, MI 1, is higher than that for the (100) orientation, M100, in the elastic regime, assuming isostrain averaging, the difference in strain energy density between a (111) and a (100) oriented grain can be written as:
AFE = (M]I - M100 )e2 = 146.4s 2 GJ/m 3 ,
(2)
where e is the biaxial strain. This driving force is independent of the sign of the applied strain. Several experimental observations support the postulate that (100) abnormal grain growth in Cu films is due to strain energy density minimization [10,23]. First, the growth is observed in films deposited under similar conditions but in chambers with base pressures varying from 10-7 to 10-10 Torr, which indicates that it is not due to impurity incorporation. In addition, (100) abnormal grain growth has been observed on P-Ta, W and amorphous C underlayers, which is suggestive that it is not due to interfacial energy minimization. Furt
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