Single crystal silicon is widely used in the fabrication of large-scale integrated circuits and photovoltaic solar cells in semiconductor industry. From the growth of the single crystal silicon to the final semiconductor devices, the silicon crystal experiences varies thermal, mechanical, physical and chemical processing, such as thermal gradients, dicing and polishing, and deposition of polyamides and metal films. All these processes introduce or change the residual stresses inside the silicon. These residual stresses may cause failure during processing or poor reliability in future service. Therefore it is critical to measure the in-situ residual stresses in the silicon in order to find the best processing parameters to control the quality of products.

  There are several methods for crystal characterization, for example X-ray topography, electron microscopy and etch-pit method. Although these methods are effectively used, they are not always convenient to determine the stress distribution in the crystal quantitatively. Though silicon is opaque to the visible light, it is transparent to near infrared radiation. Therefore near infrared (NIR) polariscope can be applied to measure the residual stresses. By applying phase stepping techniques to photoelasticity, the data can be processed automatically, and the precision can be greatly improved.

  In order to increase the sensitivity of the system, fringe multiplier is used in the infrared polariscope. The fringe multiplier is a technique used to increase the isochromatic fringe order several times as many as the ordinary technique. This multiplication is achieved by the two beam-splitters in the system. The light reflected by the two splitters passes through the specimen several times, as shown in figure 1, and the sensitivity is increased accordingly. One of the beam splitters is inclined slightly so that the light rays that have traveled through the specimen for different times emerge in different directions, and the focus points are separated on the focal plane of the focusing lens. With a spatial filter, the desired focus point that corresponds to the according fringe multiply factor can be selected for analysis. The kth focus point of forward rays represents the multiplication factor m equals to . The physical meaning of multiplication factor m is to increase the effective optical path of polariscope by m times. It is equivalent to increasing the specimen thickness m times or decreasing the illumination wavelength m times. Therefore, the sensitivity of the polariscope is increased by m times.

Polariscope arrangement and light intensities of phase stepping

  There are two main types of error in the photo stepping polariscope, namely those associated with the optical apparatus and those associated with the image process devices including camera, CCD and image digitization. For the image process devices, the digitization error of image is discussed. Error of optical apparatus in photo stepping process essentially presents itself as the angular misalignment and imperfections in the optical elements. In many cases, the system errors of the angular alignment have great influence on the light intensities. The misalignment includes the misalignment of both two waveplates and the analyzer.
  The maximum value of the total error of the misalignment is about 0.08 fringe order. The maximum error in stress is 0.5MPa when the misalignments of the first waveplate, second waveplate and analyzer are all one degree. The error is minimized in the region where the real retardation is small or isoclinic angle is zero.
  There is no apparent fringe in the whole field of interest in residual stress measurement. The main problem of the image digitization in this situation is the limited gray levels, or contrast in the six images. Due to the digitization error, the light intensity values used in equation 5 and 6 cannot be considered as exact value. The minimum digitization error usually is considered to be one gray level. The error of isochromatic parameter can be expressed by the error in intensity measurement.
  which is inverse proportional to the effective contrast of the six images, and the error of isochromatic retardation is between 0.02 and 0.04 fringe order.

  Four-point-bending tests were performed on a silicon strip diced from a (100) wafer to calibrate the system. The longitudinal direction of the silicon strip is along the [110] direction, and the transverse direction is along the . As shown in figure 7, the load is applied by two weights hung on the specimen by two strings, and the state of stress between the two loading points is pure bending. The longitudinal stress is linear along the height direction, and there are no transverse normal stress and in-plane shear stress. Therefore, the principal orientations are in the longitudinal and transverse directions, which are the [110] and orientations respectively. In this case, the accuracy can be increased by aligning the shift in the horizontal direction, along which the stress is equal, and the effect of losing spatial resolution is minimized  The residual stress in silicon wafer and WEB sample can be measured by near-infrared polariscope with fringe multiplication. Associated with the phase stepping technique, higher fringe multiplication factors can achieve higher measurement sensitivity with the loss of spatial resolution resulting from the finite dimension of the light source, as well as the tilting of the beam splitters. The spatial resolution can be improved by reducing the space between the two beam splitters and size of light source.