Hole tunneling through the emitter-base junction of a heterojunction bipolar transistor

Starting with the $4\times 4$ Luttinger-Kohn Hamiltonian, we develop a scattering-matrix approach to study coherent hole transport through the valence-band energy profile across the emitter-base junction of typical abrupt and graded $\mathrm{Pnp}$ heterojunction bipolar transistors. The tunneling and reflection coefficients of heavy and light holes are calculated for the upper and lower $2\times 2$ Hamiltonians obtained through a unitary transform of the $4\times 4$ Luttinger-Kohn Hamiltonian. The probability for a transition from heavy (light) to light (heavy) hole while tunneling across the emitter-base junction is calculated as a function of the value of the wave vector parallel to the emitter-base heterointerface for both abrupt and graded heterojunctions. For holes injected from the emitter into the base, the probability of heavy- to light-hole conversion is shown to be quite different when calculated with the upper and lower Hamiltonians. On the other hand, the probability of light- to heavy-hole conversion is nearly the same for the upper and lower Hamiltonians. The energy dependence of the heavy- and light-hole tunneling coefficients is shown to be quite different from those calculated using a parabolic-band model, in which the effects of mixing and anisotropy in the valence band are neglected.