Effect of extended confinement on the structure of edge channels in the quantum anomalous Hall effect

Abstract

Quantum anomalous Hall (QAH) effect in the films with nontrivial band structure accompanies the ferromagnetic transition in the system of magnetic dopants. Experimentally, the QAH transition manifests itself as a jump in the dependence of longitudinal resistivity on a weak external magnetic field. Microscopically, this jump originates from the emergence of a chiral edge mode on one side of the ferromagnetic transition. We study analytically the effect of an extended confinement on the structure of the edge modes. We employ the simplest model of the extended confinement in the form of potential step next to the hard wall. It is shown that, unlike the conventional quantum Hall effect, where all edge channels are chiral, in QAH effect, a complex structure of the boundary leads to nonchiral edge modes which are present on both sides of the ferromagnetic transition. Wave functions of nonchiral modes are different above and below the transition: on the "topological" side, where the chiral edge mode is supported, nonchiral modes are "repelled" from the boundary, i.e. they are much less localized than on the "trivial" side. Thus, the disorder-induced scattering into these modes will boost the extension of the chiral edge mode. The prime experimental manifestation of nonchiral modes is that, by contributing to longitudinal resistance, they smear the QAH transition.

Part-per-million quantization and current-induced breakdown of the quantum anomalous Hall effect

“…However, complications to this picture lead us to believe Zener tunneling may be less significant than the calculation suggests. If the exchange-induced gap fluctuates spatially, the energy scale corresponding to thermal activation in transport measurements may be substantially smaller than the average gap size [18]. If Δ varies strongly over a length scale smaller than λ ≈ 4 μ m, estimated from the transport gap, the true value of the onset field E 0 may be far higher due to a larger effective tunnel barrier (noting E 0 ∝ Δ 2 in a spatially uniform system).…”

“…Moreover, ρ yx deviates from quantization and ρ xx rises rapidly with temperature, typically on a scale of hundreds of millikelvin [7,12,14,15], unexpectedly small compared to the Curie temperature, which is generally tens of kelvin [3–6,8–11,16,17]. Angle-resolved photoemission spectroscopy results [17] suggest that in V-doped films, this discrepancy may be due to the bulk valence band overlapping the surface-state gap, while scanning tunneling microscopy [16] on Cr-doped bulk crystals shows strong spatial variations of the exchange-induced gap which could result in a small temperature scale in transport [18], despite an average gap size of ~30 meV. However, a comprehensive explanation of the unexpected and nonideal behavior in QAH is still lacking.…”

Current-Driven Instability of the Quantum Anomalous Hall Effect in Ferromagnetic Topological Insulators

“…Because σ xx in the transport measurement reflects the smallest gap region connecting the counterpropagating edge channels, the energy gap of the Crsparse region probably contribute to the long ξ. According to a recent theory [33], the magnetization-induced gap can survive in the presence of such Cr disorder but the gap is largely reduced compared to the case of uniform Cr distribution. We speculate that the observed activation energy T 0 may correspond to the reduced magnetizationinduced gap.…”