KT G 0wEq

where B, A, Pand n are constants, Q0 is the activation energy at zero stress, ois the applied tensile stress, k is Boltzmann's constant, T is the absolute temperature, D is the diffusion coefficient (i.e., D0 exp [QD/kT]), G is the shear modulus, b is the interatomic distance, D0 is a constant, and QD is the activation energy for self-diffusion. The values of the stress exponent n are usually in the range between 4 and 10 and are highly sensitive to the grain structure of the material. The material constant Pin Eq 1 for thin aluminum films, for example, is close to 10-27 m3/atom and about half of that in aluminum-silicon alloy films of 1 pm thickness. The activation energy at zero stress, Q0, for steady-state creep in unannealed, vapor-deposited copper films is about 1.3 eV. In sputtered 1 pm thick aluminum and aluminum-1% silicon-alloy films, Q0 is 0.56 and 1.10 eV, respectively (Ref 2). The low activation-energy value for aluminum approaches that of grain-boundary diffusion of vacancies in that metal.

In summary, the uniaxial testing method provides an excellent means for obtaining readily interpretable results in terms of tensile stress and strain. These results can be applied to well-known descriptions of the fundamental processes of plastic deformation. On the other hand, the uniaxial testing technique requires extreme care in specimen preparation and handling, and the elongations that can be obtained are often limited.

Biaxial Testing of Films. The bulge-testing technique is a biaxial testing method that can be applied to freestanding thin films. A film whose edges are fixed may be viewed as an impermeable membrane. Applying a known fluid pressure, liquid or gas, to one side of this membrane will cause it to bulge, so that the film material is strained biaxially. By monitoring the bulge expansion as a function of the fluid pressure, a stress-strain relationship is obtained. In principle, bulge testing is a straightforward technique by which problems associated with defects on the sample edges are eliminated. However, the interpretation of bulge-testing results is far more complex than that of uniaxial-test results.

Bulge testing can be performed by applying fluid pressure through a hole drilled on the back of the substrate without damaging the deposited film. An alternative procedure (Fig. 2) employs films that are removed from their substrates (Ref 3). The films are fastened by an O-ring on top of a small cylindrical chamber. A syringe pump then injects glycerol into the chamber at a constant rate. The glycerol pressure is monitored by a pressure transducer, and the injected volume is determined by the elapsed time. The bulge extension can be determined either by the displaced fluid volume or by the height of the bulge. In general, when designing bulge-testing equipment, the ratio of the bulge orifice to the film thickness should be at least between 200 and 300, so that the effect of flexural rigidity is minimized.

Films used in the bulge-testing technique must be separated from their substrates. The film-ablation techniques are identical to those described in the preceding section. It should be noted that metal films, such as aluminum or aluminum alloys, deposited on a silicon substrate tend to dissolve silicon, which often precipitates in the form of silicon-rich nodules. These nodules are removed during the ablation process together with the silicon substrate, producing very fine pinholes in the film that vitiate the bulge-testing procedure.

The analysis of the pressure-bulge deformation is generally based on a model proposed by Hill (Ref 4), which gives the meridional stress as:

where P is the fluid pressure, r0 is the radius of curvature at the top of the bulge, t is the thickness of the film, a is the radius of the orifice, and h is the height of the bulge. This is an approximate expression that is predicated on the bulge height's being much less than the radius of the orifice. The calculation of the meridional strain, also based on Hill's work, is:

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