Effects on VLSI Yield of Doubly-Stochastic Impurity Distributions

An expression is derived for the doubly-stochastic distribution of the number of impurities in the base region of a bipolar transistor; the distribution results from uncertainty in ion implantation parameters. Expressions are derived for device yield, and VLSI (very large scale integration) chip yield with an N-bit parity check. These derivations can be extended to other devices in a straightforward manner. As an example, calculations have been performed using specific parameters, which have led to the following observations: 1. The doubly stochastic effect is most sensitive to uncertainty in the straggle (standard deviation) of the emitter impurity distribution. 2. Uncertainty of the order of 5% in an implantation parameter causes substantial broadening of the distribution of impurities, for the case considered. 3. Device yield decreases rapidly for dimensions less than a well-defined threshold (≈ 0.75 μm for the case considered). 4. Chip yield, without a parity check, exhibits a threshold effect at device yield = 1-1/N
_chip
.(N
_chip
≈ number of devices per chip.) The device yield must exceed this threshold to produce large chip yields. 5. The use of a parity check reduces the device yield threshold to 1-10/N
_Chip
. Use of fewer bits per parity check reduces the threshold further. 6. For the example considered, the minimum device dimensions for large chip yields is of the order of 1 to 1.5 μm, using a 16-bit parity check. The minimum device size for reliable system performance for other cases will depend upon specific device parameters.