The viscosity of germanium during substrate relaxation upon thermal anneal

Abstract

Thin-film heterostructures experience structural relaxation when subjected to post-deposition thermal heat treatment. The rate of relaxation, elastic effects, and inelastic effects on the stress and deformation of the structure are determined by the physical properties of the materials, in particular, the solid-phase viscosity. During relaxation, movement of defects causes an increase of viscosity with time at a constant rate as these defects are annihilated. Experimental anneals have been performed on structures with polycrystalline silicon films on (111) germanium substrates, in which the substrate relaxes during thermal annealing. A numerical analysis of the experimental results has determined values for the viscosity and viscosity rate of (111) germanium wafers. In addition, four zones of the relaxation process have been identified, and results indicate that the increasing viscosity with time has a larger effect at lower furnace ramp-up rates.

Effects of viscous flow on residual stresses in film/substrate systems

“…When viscous flow occurs in the film, the resulting differential equation for stress relaxation in the film constrained by a rigid substrate has been shown, such that 21,22,24,27 …”

“…Viscous flow in film/substrate systems has been documented. Observations of viscous flow in films include that in SiO 2 /Si, 19,20 Pd-Si/Si, 21 Mo-Si/Si, 22 amorphous Si/crystalline Si, 9,10 and polyimide/Ge; 23 observations of viscous flow in substrates include that in Si/Ge 24,25 and Si 3 N 4 /Si. 26,27 In the presence of viscous flow, stress relaxation occurs and both the residual stresses and the curvature in the system are modified.…”

“…26,27 In the presence of viscous flow, stress relaxation occurs and both the residual stresses and the curvature in the system are modified. 9,10,[19][20][21][22][23][24][25][26][27] While rigorous analytical solutions have been obtained for elastic systems, 4,5,16,17 analyses for viscous flow systems are few in number, elementary, and often limited to the case of viscous flow in a thin film constrained by a rigid substrate. 21,22,24,27 In this case, the film thickness is ignored in the analyses and an analytical model describing how viscous flow in substrates relaxes residual stresses in films has not yet been developed.…”

Viscoelastic stress relaxation in film/substrate systems—Kelvin model

“…[6][7][8][9][10][11] While both the Maxwell and the Kelvin models have been developed to describe viscoelasticity, analyses of viscoelastic stress relaxation in film/substrate systems are often based on the Maxwell model. [12][13][14][15][16][17][18][19] Using the Maxwell model, solutions for stress relaxation in the system require solving partial differential equations, and the film thickness is often ignored in order to obtain closed-form solutions. [12][13][14][15][16][17][18] By using the analogy between the governing field equation of elasticity and the Laplace transform with respect to time of the viscoelastic field equation, Hsueh and Lee 19 were able to circumvent solving complex partial differential equations and to include the film thickness in the analysis.…”