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We propose a mathematical model of \textit{quantum} transistor in whichbandgap engineering corresponds to the tuning of Dirac potential in the complexfour-vector form. The transistor consists of $n$-relativistic spin qubitsmoving in \textit{classical} external electromagnetic fields. It is shown thatthe tuning of the direction of the external electromagnetic fields generatesperturbation on the potential temporally and spatially, determining the type ofquantum logic gates. The theory underlying of this scheme is on the proposal ofthe intertwining operator for Darboux transfomations on one-dimensional Diracequation amalgamating the \textit{vector-quantum gates duality} of Paulimatrices. Simultaneous transformation of qubit and energy can be accomplishedby setting the $\{\textit{control, cyclic}\}$-operators attached on thecoupling between one-qubit quantum gate: the chose of \textit{cyclic}-operatorswaps the qubit and energy simultaneously, while \textit{control}-operatorensures the energy conservation.

Dirac four-potential tunings-based quantum transistor utilizing the Lorentz force