70 Science Research Writing Th e fi lm stress, σf, can be determined by measuring the resulting substrate curvature [11], according to Stoney's formula:
σf = (Est2s/6(1 − vs)tf)(1/rs − 1/rf), (2) where rs and rf are the radii of curvature of the bare substrate and substrate with fi lm, respectively; Es, ts and vs are the Young's modulus, thickness and Poisson's ratio of the silicon substrate, respectively, and tf is the thickness of the fi lm. Tensile stresses are positive and compressive stresses negative; thus, a positive radius of curvature denotes a convex fi lm surface. Entire 75 mm diameter wafers were used, and curvature was measured from plots of surface profi le along 30 mm lines over the central part of the fi lm surface using a Dektak IIA auto-levelling profi lometer. To reduce inaccuracy caused by lack of axial symmetry in the wafer curvature, two scans were made, in orthogonal directions, for each measurement, and the inverse radii thus obtained were averaged. Care was taken not to use wafers which had a substantially asymmetric curvature before deposition. Wafer thicknesses, measured with a micrometer, were 390 ± 3 µm. Final fi lm thicknesses were measured by ellipsometry and checked by patterned etching and profi lometry, and interim thicknesses were estimated by interpolation. Equivalent single-layer thickness measurements indicate that the assumption that fi nal thickness is proportional to number of layers is suffi ciently accurate. For Es/ (1 − vs), the value 180 GPa was used [11]. In order to give an indication of the eff ect of water content on stress, 10 layers were deposited for each R value, using 10 s rapid thermal annealing at 1000°C in all cases. Infrared imaging of defects heated by a sonic pulse ii) Experiment Our experimental setup is shown in Fig. 1. Th e source of the sonic excitation is a Branson, Model 900 MA 20 kHz ultrasonic welding generator, with a Model GK-5 hand-held gun. Th e source has a maximum power of 1 kW, and is triggered to provide a
Methodology — Writing Task 71 short (typically 50–200 ms duration) output pulse to the gun. Th e gun contains a piezoelectric transducer that couples to the specimen through the 1.3-cm-diam tip of a steel horn. In the laboratory setup, as can be seen in Fig. 1, we use a mechanical fi xture to hold the sonic horn fi rmly against the sample surface. Th is setup uses a machine slide to provide reproducible alignment of the horn. Typically, a piece of soft Cu sheet is placed between the tip of the horn and the specimen to provide good sound transmission. Th e location of the source on the sample is chosen primarily for convenience of geometrical alignment, and since it has minimal eff ect on the resulting sonic IR images, typically is not changed during the course of the inspection. Sound waves at frequencies of 20 kHz in metals such as aluminium or steel have wavelengths on the order of tens of centimetres, and propagate with appreciable amplitude over distances much longer than a wavelength. For typical complex-shaped industrial parts (see, for example, the aluminium automotive part shown in Fig. 1), refl ections from various boundaries of the specimen introduce countless conversions among the vibrational modes, leading to a very complicated pattern of sound within the specimen during the time that the pulse is applied. Since the speed of sound in solids is typically on the order of a few km/s, this sound fi eld completely insonifi es the regions under inspection during the time that the excitation pulse is applied. If a subsurface interface is present, say a fatigue crack in a metal, or a delamination in a composite structure, the opposing surfaces at the interface will be caused to move by the various sound modes present there. Th e complexity of the sound is such that relative motion of these surfaces will ordinarily have components both in the plane of the crack and normal to it. Th us, the surfaces will 'rub' and 'slap' against one another, with a concomitant local dissipation of mechanical energy. Th is energy dissipation causes a temperature rise, which propagates in the material through thermal diff usion. We monitor this dissipation through its eff ect on the surface temperature distribution. Th e resolution of the resulting images depends on the depth of the dissipative source as well as on the time at which the imaging is carried out.