Disordered contacts can localize helical edge electrons

It is well known that quantum spin Hall (QSH) edge modes being helical are immune to backscattering due to non-magnetic disorder within the sample. Thus, quantum spin Hall edge modes are non-localized and show a vanishing Hall resistance along with quantized 2-terminal, longitudinal and non-local resistances even in presence of sample disorder. However, this is not the case for contact disorder. This paper shows that when all contacts are disordered in an N-terminal quantum spin Hall sample, then transport via these helical QSH edge modes can have a significant localization correction. All the resistances in an N-terminal quantum spin Hall sample deviate from their values derived while neglecting the phase acquired at disordered contacts, and this deviation is called the quantum localization correction. This correction term increases with the increase of disorderedness of contacts but decreases with the increase in the number of contacts in an N terminal sample. The presence of inelastic scattering, however, can completely destroy the quantum localization correction.