The conductivity of a two-dimensional HgTe quantum well with a width $\sim 6.3\mathrm{nm}$, close to the transition from ordinary to topological insulating phases, is studied. The Fermi level is supposed to get to the overall energy gap. The consideration is based on the percolation theory. We have found that the width fluctuations convert the system to a random mixture of domains with positive and negative energy gaps with internal edge states formed near zero gap lines. In the case with no potential fluctuations, the conductance of a finite sample is provided by a random edge states network. The zero-temperature conductivity of an infinite sample is determined by the free motion of electrons along the zero-gap lines and tunneling between them. The conductance of a single $p-n$ junction, which is crossed by the edge state, is found. The result is applied to the situation when potential fluctuations transform the system to a mixture of $p$- and $n$-domains. It is stated that the tunneling across $p-n$ junctions forbids the low-temperature conductivity of a random system, but the latter is restored due to the random edge states crossing the junctions.

• Received 21 November 2019
• Revised 29 January 2020
• Accepted 21 February 2020

DOI:https://doi.org/10.1103/PhysRevB.101.125415