Breakdown of the semi-classical conduction theory in approximants of the octagonal tiling

We present numerical calculations of quantum transport in perfect octagonal approximants. These calculations include a Boltzmann (intra-band) contribution and a non-Boltzmann (inter-band) contribution. When the unit cell size of the approximant increases, the magnitude of Boltzmann terms decreases, whereas the magnitude of non-Boltzmann terms increases. It shows that, in large approximants, the non-Boltzmann contributions should dominate the transport properties of electrons. This confirms the break-down of the Bloch-Boltzmann theory to understand the transport properties in approximants with very large unit cells, and then in quasicrystals, as found in actual Al-based approximants.