# From classical to quantum regime of topological surface states via defect engineering

### Submission summary

 As Contributors: Seongshik Oh · Xiong Yao Preprint link: scipost_202204_00022v1 Date accepted: 2022-05-04 Date submitted: 2022-04-14 00:33 Submitted by: Yao, Xiong Submitted to: SciPost Physics Lecture Notes Academic field: Physics Specialties: Condensed Matter Physics - Experiment Approach: Experimental

### Abstract

Since the notion of topological insulator (TI) was envisioned in late 2000s, topology has become a new paradigm in condensed matter physics. Realization of topology as a generic property of materials has led to numerous predictions of topological effects. Although most of the classical topological effects, directly resulting from the presence of the spin-momentum-locked topological surface states (TSS), were experimentally confirmed soon after the theoretical prediction of TIs, many topological quantum effects remained elusive for a long while. It turns out that native defects, particularly interfacial defects, have been the main culprit behind this impasse. Even after quantum regime is achieved for the bulk states, TSS still tends to remain in the classical regime due to high density of interfacial defects, which frequently donate mobile carriers due to the very nature of the topologically-protected surface states. However, with several defect engineering schemes that suppress these effects, a series of topological quantum effects have emerged including quantum anomalous Hall effect, quantum Hall effect, quantized Faraday/Kerr rotations, topological quantum phase transitions, axion insulating state, zeroth-Landau level state, etc. Here, we review how these defect engineering schemes have allowed topological surface states to pull out of the murky classical regime and reveal their elusive quantum signatures, over the past decade.

Published as SciPost Phys. Lect. Notes 58 (2022)

We would like to thank the referee for emphasizing the significance of our review paper, and for recommending publication after a minor revision.
We also would like to thank the referee for mentioning the discussion on band bending effects.
The flat band condition can be understood from the attached Figure 1. Given that the Mott criterion fixes $E_F$ to be at the conduction band minimum (confirmed by previous report) and knowing the surface state carrier density, together these fix the band bending as follows. First, assume that at the surface $E_F$ barely touches the CB minimum. Then assuming a Fermi velocity (4.0 *$10^5$ m/s) and the energy of the CB minimum relative to the Dirac point (210 meV), the surface carrier density can be calculated for one surface to be $n_{SS}$=5.0*$10^{12}$/$cm^2$ by the formula given on Page 12 and 13. Since EF at the surface and in the bulk is at the CB minimum, the bands must be flat, and band bending is negligible (see attached Figure 1). Therefore, $n_{SS}$=5.0*$10^{12}$/$cm^2$ gives the flat band condition. Either a higher or lower carrier density would give rise to an accumulation or depletion of carriers and band bending near the surface.
Due to limitations of space and scope, we are not able to extend all the details about band bending effects in topological insulators in this paper. These details about the band bending effects are included in the cited reference 84 "Matthew Brahlek et.al, Solid State Communications, 215–216, 54-62 (2015)", which is a review paper previously published by our group.
We agree with the referee that thin film growth of Weyl semimetals is important to the quantum materials community. However, the topic of the current review paper is topological insulators rather than semimetals. Discussion on Weyl semimetals are beyond the scope of this paper and should be included in another separate review paper.

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