A new bipolar transistor with an n+-GaAs/p-GaAs/sawtooth doping superlattice (SDS)/n+-GaAs structure has been demonstrated. Owing to the avalanche multiplications within SDS periods or the emitter-base p-n junction depletion region, an S-shaped negative-differential-resistance (NDR) phenomenon is observed both under the normal- and inverted-mode two-terminal operation. The bidirectional switching performances provide significant flexibility on circuit applications. Also, a transistor action with current gain of about 20 is obtained under the normal-mode three-terminal operation. Furthermore, an interesting S-shaped NDR family, controlled by the base current, is achieved at higher current regimes. However, under a distinct three-terminal-controlled operation, i.e. the inverted-mode, various transistor actions and a controllable S-shaped NDR family are performed.