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 modeling disorder with a randomly varying (complex) tunneling amplitude Γ_p=|Γ_p|exp[iϕ_p]. We show that, while for a clean junction T_p is only determined by the absolute value |Γ_p| and is independent of the phase ϕ_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, analyzing three different models for phase disorder (including correlated ones), we show that not only the amount but also the way the phase ϕ_p fluctuates determines the localization length ξ_loc and the sample-averaged transmission. Finally, we discuss the physical conditions in which these three models suitably apply to realistic cases.