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**Euler’s rotation theorem and tensor rotation technique are applied to develop a generalized mathematical model for determining photoelastic constants in arbitrary orientation of cubic crystal system. Two times rotations are utilized in the model relating to crystallographic coordinates with Cartesian coordinates. The symmetry of photoelastic constants is found to have strong dependence with rotation angle. Using the model, one can determine photoelastic constants in any orientation by selecting appropriate rotation angle. The outcome of this study helps to characterize spatial variation of residual strain in crystalline as well as polycrystalline materials having cubic structure using the experimental technique known as scanning infrared polariscope.**

The quality of semiconductor materials as well as devices is strongly dependent on residual strain induced in the materials during the growth and cooling processes [

It is well known that some mechanical and optical properties of semiconductor materials and devices strongly depend on crystal orientation [

In this study, we for the first time propose a generalized mathematical model to determine photoelastic constant in arbitrary crystal orientation combining Euler’s rotation theorem and tensor rotation technique. Using the model one can calculate different components of photoelastic constants in cubic crystals just by choosing appropriate rotation angle. Herein the model is applied to determine the orientation-dependent photoelastic constants in Si crystal as an example. However, the model developed in the present study can be used to determine photoelastic constants in cubic/Zincblende crystal structure.

The model is developed with the combination of Euler’s rotation theorem [

where the indices h, k, and l are the real integers for the case when the crystalline direction is specified in terms of the angles j and

Rotation schematics applied in the proposed model

At first, transformation of XY plane about Z axis by an angle j and then transformation of

According to Euler’s rotation theorem, the combined rotation matrix M can be given [

The photoelastic constant,

Expansion of Equation (6) yields 81 _{11}, P_{12}, and P_{44} are existed [

where

. Equivalent components of P_{ijkl} for cubic crystal system [9]

0 | 0 | 0 | |||
---|---|---|---|---|---|

0 | 0 | 0 | |||

0 | 0 | 0 | |||

0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 | |

0 | 0 | 0 | 0 | 0 |

Using the mathematical formulations derived in Equations (8) to (16), we have calculated orientation-dependent photoelastic constants from [

Figures 2(a)-(c) show a comparison among the photoelastic constants_{ }and _{11} in the [_{11}. The variations of the same components of photoelastic constants are shown in

. Experimental measurement of Photoelastic constants P_{11}, P_{12}, and P_{44} for Si crystal taken from Ref. [10]

Photoelastic constants | P_{11}_{ }Xçç<100> | P_{12}_{ }Xçç<100> | P_{44}_{ }Xçç<111> |
---|---|---|---|

−0.1053 | 0.0137 | −0.054 |

Variation of orientation-dependent photoelastic constants, , and plotted for the directions from (a) [100] to [010]; (b) [110] to [001]; (c) [100] to [001]

The variations of the photoelastic constants_{12}._{ }Similar results are found for the components

Figures 4(a)-(c) show a comparison among the orientation-dependent photoelastic constants

Variation of orientation-dependent photoelastic constants, , , , , and plotted for the directions from (a) [100] to [010]; (b) [110] to [001]; (c) [100] to [001]. The magnitude of the photoelastic constants =, =, and =

hand, the component _{44} at [

A generalized mathematical model is developed for cubic crystal system to determine photoelastic constants in arbitrary orientation with the combination of tensor rotation technique and Euler’s rotation theorem. Three independent components of photoelastic constants become nine independent components due to two times rotations. However, some of them are found symmetrical depending on the rotation direction. The magnitude and variation pattern of the photoelastic constants are also found to have direction-dependent. But, for a particular

Variation of orientation-dependent photoelastic constants, , and plotted for the directions from (a) [100] to [010], (b) [110] to [001], and (c) [100] to [001]

direction, some components are found independent of rotation angle. Here, the model is applied for silicon crystal as an example. It can be applied for any crystal having cubic/Zincblende structure.