Scaling wafer stresses and thermal processes to large wafers

Gravitational stresses limit maximum temperatures for processing semiconductor wafers. Allowable rates of heating and cooling are also limited by the combined affects of thermal and gravitational stresses, particularly in batch processing of stacked wafers. With increasing wafer diameter these limitations become more severe, requiring such measures as improved wafer support to maintain a desired processing temperature or increased wafer spacing to maintain desired rates of heating and cooling. These issues are addressed here through the derivation of scaling rules and engineering formulas relating allowable operating conditions to wafer dimensions. Gravitational stresses and displacements are first determined analytically for wafers supported by rings or at discrete points. The maximum operating temperature is then computed by comparing the calculated gravitational stresses with the temperature-dependent wafer strength. The difference between wafer strength and gravitational stress is used to determine the allowable thermal stress and allowable temperature variation across the wafer. Finally, an analytical model of radial heat transfer in a batch furnace is used to compute the maximum permissible rates of heating and cooling as a function of the instantaneous wafer temperature. Example calculations illustrate the dependence of maximum operating temperatures and allowable ramp rates on wafer dimensions, support geometry, and wafer spacing.