A two-dimensional topological insulator has a gap for bulk excitations, but conducts on its boundaries via gapless edge modes. Time-reversal symmetry prohibits elastic backscattering of electrons propagating within the edge, leading to quantized conductance at zero temperature. Inelastic backscattering, present at finite temperature, breaks the quantization and increases the edge resistance; the resistance of a long edge acquires a linear dependence on its length. A phenomenological theory that introduces the least irrelevant operators in the low-energy description predicts a strong temperature-dependence for the edge resistivity. Such a prediction is at odds with the experimentally observed weak temperature dependence of the resistance.
We attempt to resolve the issue by studying a realistic microscopic mechanism for inelastic backscattering. The small band gaps in the existing putative two-dimensional topological insulators make them sensitive to static charge fluctuations created by randomness in the density of dopant. Charge fluctuations may lead to the formation of electron and hole puddles. Such a puddle – a quantum dot – tunnel-coupled to the edge may significantly enhance the inelastic backscattering rate. The added resistance is especially strong for dots carrying an odd number of electrons, due to the Kondo effect. For the same reason, the temperature dependence of the added resistance becomes rather weak. We present a detailed theory of the puddles’ effect on the helical edge properties, including (i) linear conductance, (ii) non-linear $I$–$V$ characteristic, and (iii) current noise.