Thermally induced effects during initial stage of crystal growth from melts

A Semianalytical Thermal Stress Model for the Czochralski Growth of Type III-V Compounds

“…In the fully unsteady case, the asymptotic expansion results in a system of onedimensional equations and the thermal stress can be obtained explicitly in an analytical form, under the plane strain assumption. In the pseudosteady limit this reduces to the classical result that the stress is proportional to the concavity of the temperature field [22,34]. This also extends the work of [19], where stress was obtained for a cylindrical crystal with a flat crystal-melt interface.…”

“…If the γ, δ pair is chosen above the curve S 0 (t) = 0 γ > γ max (δ) = k 2 (λ 0 + δ)/(k 2 + δλ 0 ), λ 0 = k tanh kZ 0 , Θ ch = 0 , the seed melts back. If we are below the curve, then S 0 (t) increases without bound and the curve asymptotically approaches γ = k. For small S 0 , (22) gives…”

“…To the left is the evolution of the length of a cylindrical crystal grown using the pseudosteady approximation according to (22). For the simulations, Θ ch = 0, R 0 = 1, Z 0 = 0.162 (3 cm), and = 0.026 (hgs = 4W/m 2 K).…”

Thermal stress inhibition in double crucible Czochralski large diameter crystal growth

“…A detailed theoretical analysis originally performed by Indenbom [12,13] and supported by good agreement with numerical simulations [14][15][16] suggests that the thermal stress is defined by the second-order derivative of the temperature as s st ffi aEW 2 d 2 T=dy 2 ;…”

“…In effect, in the general case of a crystal growing from the melt, the closer to the solidification interface the higher the temperature and the higher the value of q 2 T=qy 2 ; and thus the higher the value of thermal stress, which follows from the onedimensional (1D) approximation of the temperature distribution valid for low diameter crystals [16] …”