  We provide a transmission line representation for channels exhibiting
spin-momentum locking (SML) which can be used for both time-dependent and
steady-state transport analysis on a wide variety of materials with spin-orbit
coupling such as topological insulators, heavy metals, oxide interfaces, and
narrow bandgap semiconductors. This model is based on a time-dependent
four-component diffusion equation obtained from the Boltzmann transport
equation assuming linear response and elastic scattering in the channel. We
classify all electronic states in the channel into four groups ($U^+$, $D^+$,
$U^-$, and $D^-$) depending on the spin index (up ($U$), down ($D$)) and the
sign of the $x$-component of the group velocity ($+,-$) and assign an average
electrochemical potential to each of the four groups to obtain the
four-component diffusion equation. For normal metal channels, the model
decouples into the well-known transmission line model for charge and a
time-dependent version of Valet-Fert equation for spin. We first show that in
the steady-state limit our model leads to simple expressions for charge-spin
interconversion in SML channels in good agreement with existing experimental
data on diverse materials. We then use the full time-dependent model to study
spin-charge separation in the presence of SML, a subject that has been
controversial in the past. Our model shows that the charge and spin signals
travel with two distinct velocities resulting in well-known spin-charge
separation which is expected to persist even in the presence of SML. However,
our model predicts that the lower velocity signal is purely spin while the
higher velocity signal is largely charge with an additional spin component
which has not been noted before. Finally, we note that our model can be used
within standard circuit simulators like SPICE to obtain numerical results for
complex geometries.
