Analytic solutions of the effective-mass equation in strained Si-${\mathrm{Si}}_{1\mathrm{-}\mathrm{x}}$${\mathrm{Ge}}_{\mathrm{x}}$ heterostructures applied to resonant tunneling

The tunnel current in the valence band of Si-${\mathrm{Si}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Ge}}_{\mathrm{x}}$ double-barrier structures is calculated within the framework of the three-band effective-mass equation including the spin-orbit split-off band and the built-in strain. We present analytic solutions of the effective-mass equation in each bulk constituent and use them to calculate the transmission coefficient and tunnel current via a transfer-matrix technique. Though the tunnel current originates only from incoming heavy-hole states, these are strongly mixed into light-hole (LH) and spin-orbit split-off (SO) states, and several resonances in the tunnel current are even dominated by outgoing LH or SO states. An estimate is done to compare the calculated resonant bias values with experimental results.