Interface effects on the magnetic-proximity-induced quantized Hall response in heterostructures based on three-dimensional topological insulators

V. N. Men'shov, I. A. Shvets, and E. V. Chulkov

Phys. Rev. B 99, 115301 – Published 4 March 2019

Abstract

We report a theoretical study of the spin-dependent transport properties in heterostructures containing a three-dimensional topological insulator (TI) thin film and ferromagnetic normal insulator (FMNI) slab. Within the framework of a continual approach for the FMNI/TI/FMNI trilayer model, we reveal how the magnetic proximity effect at the TI/FMNI interface can influence an intrinsic Hall response of the system. We predict that the FMNI/TI/FMNI trilayer undergoes a transition into the quantum anomalous Hall phase either from the topologically trivial phase or from the quantum spin Hall phase, which is controlled by tuning the proximity-induced exchange field, the TI film thickness, and the TI band structure parameters. We draw the corresponding phase diagram of the FMNI/TI/FMNI trilayer. Moreover, we argue that the roughness at the TI/FMNI interfaces can cause the decomposition of the TI film into topologically distinct domains, which affects the Hall conductivity. We discuss the specifics of manifestation and complexities of observation of quantized conductivity in realistic TI/FMNI heterostructures.

Received 17 October 2018

DOI:https://doi.org/10.1103/PhysRevB.99.115301

Physics Subject Headings (PhySH)

Edge states Hall effect

Interfaces Magnetic semiconductors Nanostructures Narrow band gap systems Topological insulators

Mean field theory k dot p method

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. N. Men'shov^1,2,3,4, I. A. Shvets^3,4, and E. V. Chulkov^1,4,5

¹Donostia International Physics Center (DIPC), P. de Manuel Lardizabal 4, 20018 San Sebastián, Basque Country, Spain
²NRC Kurchatov Institute, Kurchatov Sqr. 1, 123182 Moscow, Russia
³Tomsk State University, 634050 Tomsk, Russia
⁴Saint Petersburg State University, Saint Petersburg 198504, Russia
⁵Departamento de Física de Materiales UPV/EHU, CFM-MPC UPV/EHU, 20080 San Sebastián/Donostia, Basque Country, Spain

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Vol. 99, Iss. 11 — 15 March 2019

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Images

Figure 1
Scheme of the FMNI/TI/FMNI trilayer. The TI film, indicated by yellow color, is sandwiched by the FMNI slabs, indicated by cyan color. The FM order in the FMNI slabs is symbolized by the set of the vertical blue arrows oriented along the $z$ quantization axis, which is perpendicular to the interface plane. The interface potential $U$ is localized at the TI/FMNI interfaces, which are highlighted in pink. The side surfaces of the film, highlighted in orange, host the edge potential $V$ . In the QAHE regime, electrons propagate along the side surfaces.
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Figure 2
Topological phase diagram of the FMNI/TI/FMNI trilayer in terms of the proximity-induced exchange splitting $ω$ and the TI film thickness $l$ under strong potential $U_{p}$ when $λ = 0.1$ . Here, $\tilde{l} = l \sqrt{\frac{Ξ_{0}}{B}}, \tilde{ω} = \frac{ω \sqrt{B}}{2 | A | \sqrt{Ξ_{0}}}$ , where $ω$ is given by Eq. (20). One can observe three topologically distinct insulating phases characterized by the quantized spin Chern number and quantized charge Chern number and represented by the corresponding regions. White regions mark the trivial phase with $C^{S} = C^{C} = 0$ . Green regions mark QSHE phase with $| C^{S} | = 1, C^{C} = 0$ . Yellow regions mark the mixed QAHE+QSHE phase with $| C^{S} | = \frac{1}{2}, | C^{C} | = \frac{1}{2}$ . Boundaries selecting different phases are given by equations $Δ_{0}^{Σ} (l, ω) = 0$ and $b^{Σ} (l, ω) = 0$ . The triangles and circles (hollow and solid) mark the points in the parameter space, at which we below compute the electron spectrum and the Berry curvature.
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Figure 3
Evolution of the dispersion relation $E_{\pm}^{Σ} (κ)$ (upper row) and the Berry curvature $Ω^{Σ} (κ)$ (lower row) with the TI film thickness [(a) for $\tilde{l} = 2.5 ... 4.0$ and (b) for $\tilde{l} = 4.5 ... 6.0]$ taking place in the FMNI/TI/FMNI trilayer. Here, $\tilde{E} = \frac{E}{Ξ_{0}}, \tilde{Ω} = Ω \frac{Ξ_{0}}{B}, \tilde{k} = k_{x, y} \sqrt{\frac{B}{Ξ_{0}}}, \tilde{l} = l \sqrt{\frac{Ξ_{0}}{B}}$ . In order to provide a comprehensive picture of the evolution, we choose eight different thicknesses denoted by solid circles at the finite value of $\tilde{ω}$ and by hollow circles in absence of the exchange splitting in the phase diagram in Fig. 2. The left part of every plot relates to $\tilde{ω} = 0.0$ , while the right part relates to $\tilde{ω} = 0.1$ . The dispersion of the four lowest subbands near the point $κ = 0, E_{\pm}^{Σ} (κ)$ and the momentum distribution of the Berry curvature for opposite quasispin polarizations $Ω^{Σ} (κ)$ are drawn in the corresponding plots on the background with color associated with a quantum phase in the diagram in Fig. 2. The curves are computed taking the same parameters as were used in Fig. 2.
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Figure 4
Evolution of the dispersion relation $E_{\pm}^{Σ} (κ)$ (upper row) and the Berry curvature $Ω^{Σ} (κ)$ (lower row) with the exchange splitting taking place in the FMNI/TI/FMNI trilayer. Here, $\tilde{E} = \frac{E}{Ξ_{0}}, \tilde{Ω} = Ω \frac{Ξ_{0}}{B}, \tilde{k} = k_{x, y} \sqrt{\frac{B}{Ξ_{0}}}, \tilde{ω} = \frac{ω \sqrt{B}}{2 | A | \sqrt{Ξ_{0}}}$ . In order to provide a comprehensive picture of the evolution, we choose five different values of the exchange splitting denoted by triangles at the fixed thickness $\tilde{l} = 5.5$ in the phase diagram in Fig. 2. The dispersion of the lowest four subbands near the point $κ = 0, E_{\pm}^{Σ} (κ)$ and the momentum distribution of the Berry curvature for opposite quasispin polarizations $Ω^{Σ} (κ)$ are drawn in the corresponding plots on the background with color associated with a quantum phase in the diagram in Fig. 2. The curves are computed taking the same parameters as were used in Fig. 2.
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Figure 5
Schematic illustration of domain structure of the TI thin film sandwiched between the FMNI slabs. The interface morphology is represented by terraces separated by QL steps oriented along hexagonal axes consistently with the threefold symmetry of tetradymite semiconductors. The sketch shows a fragment of the trilayer patterned into the Hall bar geometry (see the inset), which includes the current drain contact and one of probes. The interface roughness in the TI film is expected to induce domains of topologically different phases along the interface plane $(x, y)$ , which are separated by the domain walls hosting inner conducting channels. Topography of domain structure can be represented by islands (labeled by “ID'), regions adjoining to side surface (labeled by “AD”) and macrodomains (labeled by “MD”). See text for details. Here, one observes a particular configuration consisting of three topologically distinct insulating phases: the trivial phase (white regions); QSHE phase (green regions); the mixed QAHE+QSHE phase (yellow regions). For instance, in accordance with the phase diagram in Fig. 2, this configuration might occur in the TI film with the thickness fluctuating in the range of $2.5 ≲ \tilde{l} ≲ 4$ and $\tilde{ω} ≃ 0.15$ . Outer edge channel appearing along the side surface is indicated by the thick red line.
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