The problem of determining the doping profile minimizing base transit time TB in a BJT is investigated for the case in which the net active dopant dose Q in the quasi-neutral base is held fixed, but the width W of the base is allowed to vary. It is found that the minimum TB is obtained, for reasonable values of Q, by shrinking the base width until the profile becomes box-shaped with doping equal to the solid solubility. This occurs at so narrow a base width, however, that in any practical device it will always be favourable to grade the base doping profile.

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Journal Solid State Electronics
Winterton, S.S., & Tarr, N.G. (1998). Base transit time minimization with fixed base dopant dose. Solid State Electronics, 42(4), 667–670.