PSI - Issue 42

Neha Duhan et al. / Procedia Structural Integrity 42 (2022) 863–870 Duhan et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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Joule heat along the dislocation line. The domain used for the analysis purpose has a square shape with a size L around 100 times the size of the lattice constant ( a ). The perfect edge dislocation is modelled according to the Volterra concept. The location of the dislocation within the L size square domain is at the exact center of the domain. The Burgers vector is a function of the lattice vector taken as   1 110 2 = b with magnitude as 2 a b = . The lattice constant ( a ) of the alloy is a function of the amount of silicon mole fraction ‘x Si ’ , with an expression slightly different from Vegard’s Law (Dismukes et al. (1964b)) is given as follows, ( ) ( ) 2 Si Si 5.431 0.2 1 x 0.027 1 x a = + − + − Å (17) The thermal expansion coefficient for silicon-germanium alloy in terms of the amount of germanium mole fraction ‘x Ge ’ , which is equal to ‘1 - x Si ’ is given by Levinshtein et al. (2001) as follows, ( ) ( ) 6 Ge Ge 6 Ge Ge 2.6 2.55x 10 for x 0.85 7.53x 0.89 10 for x 0.85  − − +   = −   1/K (18) The thermal conductivity as a function of ‘x Ge ’ is given as follows ( Levinshtein et al. (2001)), ( ) Ge Ge 0.046 0.084x for 0.2 x 0.85  = +   W/cm-K (19) For values of ‘ x Ge ’ not included by Eq. (19), the thermal conductivity is taken from the thermal resistivity plot given in Kasper and Lyutovich (2002). Furthermore, the electrical conductivity of Si x Ge 1-x alloy is taken from the plot of intrinsic conductivity vs. ‘x Si ’ available in the literature (Madelung (2012)). Other than this, the elastic constants of the Si x Ge 1-x alloy are obtained from the rule of mixture as follows (Kasper and Lyutovich (2002)), ( ) ,Ge Ge ,Si Ge x 1 x ij ij ij c c c = + − GPa (20) The values of elastic constants c 11 , c 12 and c 44 individually for Si and Ge are given in Table 1.

Table 1. Elastic Material Properties of Si and Ge (Madelung (2012)). Property Si

Ge

c 11 (GPa) c 12 (GPa) c 44 (GPa)

165.77

124 41.3 68.3

63.93 79.62

The Joule heat is computed for an electric field of 1.3 kV/cm acting along the dislocation line in the z-direction. The temperature difference changes between 300 K and 500 K, which generates the heat flux along and normal to the glide plane of dislocation. Different Si x Ge 1-x alloys are considered for the analysis by changing the concentration of Si between 0 and 1 in the alloy. Fig. 2(a) shows the schematic of the semi-infinite Si x Ge 1-x alloy domain with an edge dislocation at a distance d = L /2 from the left surface and L /2 above the bottom surface of the domain. The boundary conditions as the displacement field of the edge dislocation in the semi-infinite domain (Oswald et al. (2011)) are applied on the outer boundary of the problem domain (orange color). The left side surface is the free surface of the problem domain hence no displacement boundary conditions. The heat flux normal to the glide plane is applied by defining the temperature difference on the top (positive) and bottom (negative) surface shown by blue color lines. For the case of heat flux along the glide plane, the temperature difference is specified on the right (positive) and left (negative) surface shown by green color lines. Fig. 2(b) shows the mesh used for solving the XFEM problem of the edge dislocation in alloy semiconductor. The nodes shown by red color circles are outer boundary nodes on which different values of displacements and temperatures are specified. The element nodes at the dislocation core are specified with magenta * and element nodes along the glide plane are specified with blue squares. The number of elements used are 3481, which are obtained for sufficiently good results with error in an acceptable range.