Effects of disorder on electron tunneling through helical edge states

A tunnel junction between helical edge states, realized via a constriction in a Quantum Spin Hall system, can be exploited to steer both charge and spin current into various terminals. We investigate the effects of disorder on the transmission coefficient $T_p$ of the junction, by modelling disorder with a randomly varying (complex) tunneling amplitude $\Gamma_p=|\Gamma_p| \exp[i \phi_p]$. We show that, while for a clean junction $T_p$ is only determined by the absolute value $|\Gamma_p|$ and is independent of the phase $\phi_p$, the situation can be quite different in the presence of disorder: phase fluctuations may dramatically affect the energy dependence of $T_p$ of any single sample. Furthermore, analysing three different models for phase disorder (including correlated ones), we show that not only the amount but also the way the phase $\phi_p$ fluctuates determines the localisation length $\xi_{loc}$ and the sample-averaged transmission. Finally, we discuss the physical conditions in which these three models suitably apply to realistic cases.