Exact solution for space-charge broadened packets in semiconductors

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Abstract
The transport equation for charge packets in time-of-flight measurements of drift velocity in bulk semiconductors including full account of space-charge effects is derived. It is found to be a nonlinear, partial, integro-differential equation. Two key transformations permit finding an exact, analytical solution by decomposing the problem into separate equations for the shape function, the broadening function, and the velocity function. The solution consists of a rectangular-shaped charge packet which broadens linearly in time and propagates with an exponentially increasing velocity until it begins entering the electrode at which time the velocity can be expressed in terms of modified Bessel functions. The observed current pulse in the external circuit is also calculated and it is shown that experimenters have made a mistake in interpreting the pulse and so have deduced too high a drift velocity. The error, however, has typically been small (∼1%).