Robustness of quantized transport through edge states of finite length: Imaging current density in Floquet topological versus quantum spin and anomalous Hall insulators

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Robustness of quantized transport through edge states of finite length: Imaging current density in Floquet topological versus quantum spin and anomalous Hall insulators

Utkarsh Bajpai, Mark J. H. Ku, and Branislav K. Nikolić

Phys. Rev. Research 2, 033438 – Published 17 September 2020

Abstract

The theoretical analysis of topological insulators (TIs) has been traditionally focused on infinite homogeneous crystals with band gap in the bulk and nontrivial topology of their wave functions, or infinite wires whose boundaries host surface or edge metallic states. Such infinite-length edge states exhibit quantized conductance which is insensitive to edge disorder, as long as it does not break the underlying symmetry or introduce energy scale larger than the bulk gap. However, experimental devices contain finite-size topological region attached to normal metal (NM) leads, which poses a question about how precise is quantization of longitudinal conductance and how electrons transition from topologically trivial NM leads into the edge states. This particularly pressing issue for recently conjectured two-dimensional (2D) Floquet TI where electrons flow from time-independent NM leads into time-dependent edge states, the very recent experimental realization [J. W. McIver et al., Nat. Phys. 16, 38 (2020)] of Floquet TI using graphene irradiated by circularly polarized light did not exhibit either quantized longitudinal or Hall conductance. Here, we employ a charge-conserving solution for Floquet-nonequilibrium Green functions of irradiated graphene nanoribbon to compute longitudinal two-terminal conductance, as well as spatial profiles of local current density as electrons propagate from NM leads into the Floquet TI. For comparison, we also compute conductance of graphene-based realization of 2D quantum Hall, quantum anomalous Hall, and quantum spin Hall insulators. Although zero-temperature conductance within the gap of these three conventional time-independent 2D TIs of finite length exhibits small oscillations due to reflections at the NM-lead/2D-TI interface, it remains very close to perfectly quantized plateau at $2 e^{2} / h$ and completely insensitive to edge disorder. This is due to the fact that inside conventional TIs there is only edge local current density which circumvents any disorder. In contrast, in the case of Floquet TI both bulk and edge local current densities contribute equally to total current, which leads to longitudinal conductance below the expected quantized plateau that is further reduced by edge vacancies. We propose two experimental schemes to detect coexistence of bulk and edge current densities within Floquet TI: (i) drilling a nanopore in the interior of irradiated region of graphene will induce backscattering of bulk current density, thereby reducing longitudinal conductance by $\sim 28 %$ ; (ii) imaging of magnetic field produced by local current density using diamond nitrogen-vacancy centers.

Received 30 June 2020
Revised 13 August 2020
Accepted 27 August 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033438

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Edge states Integer quantum Hall effect Quantum anomalous Hall effect Quantum spin Hall effect Quantum transport

Floquet systems

Nonequilibrium Green's function

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Utkarsh Bajpai¹, Mark J. H. Ku ^1,2, and Branislav K. Nikolić ^1,*

¹Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA
²Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, USA

^*bnikolic@udel.edu

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Vol. 2, Iss. 3 — September - November 2020

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Images

Figure 1
Schematic view of a two-terminal device where an infinite ZGNR is attached to two macroscopic reservoirs with chemical potentials $μ_{L}$ and $μ_{R}$ on the left and right, respectively, so that $μ_{L} - μ_{R} = e V_{b}$ is externally applied dc bias voltage. In (a), the scattering region (blue shaded) in the middle of finite length $L = 30 \sqrt{3} a$ and width $W = 29 a$ is Floquet TI generated by irradiating [31] segment of ZGNR by circularly polarized light of intensity $z$ and frequency $ω$ [Eq. (2)]. In (b), the scattering region (shaded green) is quantum Hall, quantum anomalous Hall, or quantum spin Hall insulator with their parameters tuned to produce the same topologically nontrivial band gap $Δ_{g}$ [Fig. 2] in the bulk of all such conventional time-independent TIs.
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Figure 2
(a) The zero-temperature two-terminal conductance vs the Fermi energy $E_{F}$ of device in Fig. 1 where central scattering region hosts finite-length conventional 2D time-independent TIs, such as QHI, QAHI, and QSHI. The TIs are defined on pristine or edge disordered (denoted by ED) ZGNR due to vacancies illustrated in Figs. 4, 4, and 4. The zoom-in of conductance values within the rectangle in (a) is shown in (b)–(d) for pristine ZGNR, and (e)–(g) for edge-disordered ZGNR. The two NM leads, from which electrons are injected into the topologically protected edge states of finite length with the corresponding local current density profiles shown in Figs. 4–4, are also made of ZGNR of the same width as the scattering region [Fig. 1]. The gap in the bulk of all three 2D TIs is tuned to $Δ_{g} = 0.54 γ$ and marked in (a).
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Figure 3
(a) Quasienergy spectrum $ξ_{QE} (k_{x})$ for an infinite ZGNR that is irradiated by circularly polarized monochromatic laser light of frequency $ℏ ω = 3 γ$ and intensity $z = 0.5$ over its whole length. The spectrum is obtained by diagonalizing the corresponding Floquet Hamiltonian [Eq. (7)] truncated to $- N_{ph} < n < N_{ph}$ Floquet replicas where $N_{ph} = 7$ is chosen. The yellow shaded region marks the topological gap $Δ_{0}$ around $ξ_{QE} = 0$ corresponding to CNP, while the red shaded region marks the dynamical topological gap $Δ_{1}$ around $ξ_{QE} = \pm ℏ ω / 2$ . (b) The zero-temperature two-terminal conductance vs $E_{F}$ (computed using $N_{ph} = 7$ ) of two-terminal device in Fig. 1 whose scattering region is Floquet TI of finite length due to irradiation by circularly polarized light. The pristine irradiated ZGNR is marked by FTI and irradiated edge-disordered ZGNR is marked by FTI-ED. The conductance of an infinite nonirradiated (NIR) pristine ZGNR is also shown as a reference. (c) Total DOS for the same device marked by FTI in (b). (d) Convergence of lead currents $I_{L}$ and $I_{R}$ vs $N_{ph}$ at $E_{F} = ℏ ω / 2$ .
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Figure 4
Spatial profiles of local current density in two-terminal devices of Fig. 1 where the scattering region (dotted rectangle) of finite length is (a) irradiated pristine ZGNR hosting Floquet TI; (b) irradiated edge-disordered ZGNR; (c) pristine QHI; (d) edge-disordered QHI; (e) pristine QAHI; (f) edge-disordered QAHI; (g) pristine QSHI; and (h) edge-disordered QSHI. In (a) and (b) we use $ℏ ω = 3 γ$ , $z = 0.5$ , $N_{ph} = 7$ , and $E_{F} = ℏ ω / 2$ corresponding to the middle of $Δ_{1}$ gap in Fig. 3. In (c)–(h), $E_{F} = 0.2 γ$ , and in (g) and (h) $t_{SO} = 0.1 γ$ . The black solid arrows are guide to the eye to indicate the spatial region with large flux of local current density.
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Figure 5
Spatial profile of local current density in Figs. 4 and 4 over the transverse cross section within: (a) pristine Floquet TI or (b) Floquet TI with edge disorder. The position of the transverse cross section is marked by dashed vertical line in Figs. 4 and 4, respectively. The horizontal dashed line in both panels marks the extent of the edge state.
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Figure 6
(a) The LDOS [Eq. (25)] evaluated at $E = ℏ ω / 2$ in the center of $Δ_{1}$ gap in Fig. 3 for irradiated pristine ZGNR. (b) The LDOS evaluated at $E = ℏ ω / 2$ for irradiated ZGNR with a nanopore drilled in the interior of the nanoribbon. (c) Time-averaged local bond current [Eq. (19)] in the same irradiated ZGNR with a nanopore as in (b). (d) Zero-temperature two-terminal conductance $G (E_{F})$ of irradiated ZGNR with a nanopore (orange line) vs conductance of irradiated pristine ZGNR (blue line) within the gap $Δ_{1}$ in Fig. 3. The former is reduced by $\sim 28 %$ with respect to the latter. In all panels we use $ℏ ω = 3 γ$ , $z = 0.5$ , and $N_{ph} = 7$ .
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