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 Gamma(p) = vertical bar Gamma(p)vertical bar exp[i phi(p)]. We show that, while for a clean junction T-p is only determined by the absolute value vertical bar Gamma(p)vertical bar 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, analyzing 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 localization length.loc and the sample-averaged transmission. Finally, we discuss the physical conditions in which these three models suitably apply to realistic cases.