Verification of the integral charge-control relation for high-speed bipolar transistors at high current densities

Abstract

Gummel's integral charge-control relation (ICCR) I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</inf> = (const/Q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> ).exp (V <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BE</inf> /V <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> ) is an important basis for developing self-consistent compact transistor models for the high-current region (including quasi-saturation). Such models are required for the simulation of future high-speed IC's with a high integration level. Unfortunately, the simplifying assumptions on which the ICCR is based seem to be doubtful especially for very fast transistors. Therefore, in this paper, the ICCR and its assumptions are checked via numerical simulation of such transistors (f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> ≈ 7-8 GHz). It is found that the one-dimensional ICCR is a fairly good approximation far into the high-current region. This satisfactory result is mainly due to the partial compensation of the influences of the spatially dependent doping concentration on both the electron mobility µ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> and the effective intrinsic density n <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ie</inf> within the product µ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ie</inf> . Only in the emitter and in the emitter-base space-charge region there is a strong increase of this product which, in conjunction with the increasing contribution of the hole charge in these regions, was proved to be responsible for the errors observed at high current levels. The ICCR can also be applied to a two-dimensional transistor by additionally taking into account the excess hole charge stored outside the internal transistor for the determination of Q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> . Thus the contribution of the minority charges can still be determined experimentally by measuring τ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</inf> (I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</inf> ).