Topological current divider

Francesco Romeo

Phys. Rev. B 102, 195427 – Published 23 November 2020

Abstract

We study the transport properties of a hybrid junction made of a ferromagnetic lead in electrical connection with the helical edge modes of a two-dimensional topological insulator. In this system, the time-reversal symmetry, which characterizes the ballistic edge modes of the topological insulator, is explicitly broken inside the ferromagnetic region. This conflict situation generates unusual transport phenomena at the interface which are the manifestation of the interplay between the spin polarization of the injected current and the spin-momentum locking mechanism operating inside the topological insulator. We show that the spin-polarized current originated in the ferromagnetic region is asymmetrically divided in spatially separated branch currents sustained by edge channels with different helicity inside the topological insulator. The above findings provide the working principle of a topological current divider in which the relative intensity of the branch currents is determined by the polarization of the incoming current. We discuss the relevance of this effect in spintronics where, for instance, it offers an alternative way to measure the current polarization generated by a ferromagnetic electrode.

Received 26 August 2020
Accepted 9 November 2020

DOI:https://doi.org/10.1103/PhysRevB.102.195427

Physics Subject Headings (PhySH)

Edge states Magnetism Magnetotransport Spintronics Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Francesco Romeo

Dipartimento di Fisica “E. R. Caianiello,” Università degli Studi di Salerno, Via Giovanni Paolo II, I-84084 Fisciano (Sa), Italy

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Vol. 102, Iss. 19 — 15 November 2020

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Images

Figure 1
(a) Schematic of the topological current divider described in the main text. A two-dimensional topological insulator (TI) is laterally etched to form a constriction. A ferromagnetic electrode (F) is created in the constriction region where massive electronic states belonging to the F region hybridize with the massless helical modes of the topological insulator. The application of a voltage bias to the system induces an electrochemical potential gradient ( $μ_{L} \neq μ_{R}$ ) responsible for a current. The current coming from the ferromagnetic region is split in two branch currents: the top current sustained by edge modes with positive helicity ( $R ↑$ and $L ↓$ states), and the bottom current sustained by edge modes with negative helicity ( $R ↓$ and $L ↑$ states). The top and bottom currents have different intensity, this difference being controlled by the current polarization instead of the electrical resistances of the branches. (b) Side view of the device. (c) Band structure of the ferromagnetic region and of the topological channel. The Fermi level, which crosses both the spin-up and the spin-down bands of the ferromagnet, lies inside the bulk gap separating the conduction band (CB) and the valence band (VB) of the topological insulator.
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Figure 2
Schematic of the device effective model. Modes labeled by 1 and 2 belong to the ferromagnetic electrode and represent the spin-up and spin-down channels, respectively. Modes labeled by 3 and 4 belong to the topological insulator and represent positive- and negative-helicity edge states, respectively.
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Figure 3
Interface model explained in the main text. An interface potential proportional to the Dirac delta function is introduced at $x = 0$ , while a velocity gradient is present at $x = ε$ . When the limit $ε \to 0^{+}$ is considered, the matching matrix of the resulting interface $M = M_{2} M_{1}$ is recognized.
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Figure 4
(a) $P_{c}$ as a function of $h_{e x} / E_{F}$ obtained by setting the model parameter as $m_{\pm} = 1$ , $g_{0} = 2.5$ , $g_{1} = 0.5$ , $λ = 5$ (dashed line), or $λ = 2$ (solid line). (b) $P_{c}$ as a function of $h_{e x} / E_{F}$ obtained by setting $m_{+} = 3$ and $m_{-} = 1$ , while taking the remaining parameters as in (a). (c) Density plot of $P_{c}$ as a function of $h_{e x} / E_{F}$ and $m_{+}$ . The remaining model parameters have been fixed as $m_{-} = 1$ , $g_{0} = 2.5$ , $g_{1} = 0.5$ , and $λ = 2$ .
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Figure 5
(a) $J_{T}^{c} / J_{B}^{c}$ as a function of $h_{e x} / E_{F}$ and $m_{+}$ obtained by setting the model parameters as $m_{-} = 1$ , $g_{0} = 2.5$ , $g_{1} = 0.5$ , and $λ = 2$ (metallic regime). (b) $J_{T}^{c} / J_{B}^{c}$ as a function of $h_{e x} / E_{F}$ and $m_{+}$ obtained by setting the model parameters as $m_{-} = 1$ , $g_{0} = 9$ , $g_{1} = 0.5$ , and $λ = 2$ (tunneling regime). (c) $J_{T}^{c} / J_{B}^{c}$ as a function of $h_{e x} / E_{F}$ in the metallic regime. Different curves are obtained by setting $m_{+} = 0.5$ (top curve), $m_{+} = 1$ (middle curve), and $m_{+} = 1.5$ (bottom curve). (d) $J_{T}^{c} / J_{B}^{c}$ as a function of $h_{e x} / E_{F}$ in the tunneling regime. Different curves are obtained by setting $m_{+} = 0.5$ (top curve), $m_{+} = 1$ (middle curve), and $m_{+} = 1.5$ (bottom curve).
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