Tunable geometric phase of Dirac fermions in a topological junction

Sang-Jun Choi, Sunghun Park, and H.-S. Sim

Phys. Rev. B 87, 165420 – Published 10 April 2013

Abstract

We predict a tunable and nonadiabatic Berry phase effect of Dirac fermions, which is an electronic analog of the Pancharatnam phase of polarized light. The Berry phase occurs as a scattering phase shift in a single scattering event of transmission or reflection of Dirac fermions at a junction with a spatially nonuniform mass gap, unveiling the topological aspects of scattering of chiral Dirac fermions. This geometric phase plays different roles in solids as compared with the Pancharatnam phase in optics. It provides a unique approach of detecting the Chern number of the insulator side in a metal-insulator junction of Dirac fermions, implying a different type of bulk-edge correspondence at the boundary between a metal and an insulator. This phase also modifies the quantization rule of Dirac fermions, suggesting geometric-phase devices with nontrivial charge and spin transport such as a topological waveguide and a topological transistor.

Received 14 September 2012

DOI:https://doi.org/10.1103/PhysRevB.87.165420

Authors & Affiliations

Sang-Jun Choi, Sunghun Park^*, and H.-S. Sim^†

Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea

^*lngch4@kaist.ac.kr
^†hssim@kaist.ac.kr

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Vol. 87, Iss. 16 — 15 April 2013

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Figure 1
Pancharatnam phase of light and Dirac fermions. (a) Light with initial polarization $Z_{1}$ acquires Pancharatnam phase $- Ω_{Z_{1} Z_{2} Z_{3}} / 2$ after passing through polarizers with polarization axes $Z_{2}$ , $Z_{3}$ , and $Z_{1}$ . (b) Dirac fermions acquire Pancharatnam phase $P_{I \bar{T} R} = - Ω_{I \bar{T} R} / 2$ in a scattering event at a 2D step junction with mass gap. $χ_{I}$ , $χ_{R}$ , $χ_{T}$ , and $χ_{\bar{T}}$ denote the spin states involved in the scattering. $Ω_{A B C}$ is the solid angle of the geodesic polygon on the Poincaré (Bloch) sphere which sequentially connects the vertices representing polarization (spin) states $A$ , $B$ , $C$ (see arrows).Reuse & Permissions
Figure 2
Spin of evanescent states and Chern number. (a) Spin direction (arrows) of evanescent states in an edge of an insulator ( $x y$ plane), whose energy and momentum tangential to the edge are denoted as $ε$ and $k_{∥}$ , respectively; its $x y$ ( $z$ ) component is drawn parallel to the $k_{∥}$ ( $ε$ ) axis and depends on the Chern number $C$ of the insulator energy band (shade). When $C = - 1$ ( $C = 0$ ), the $x y$ spin component aligns parallel (antiparallel) to the edge (see the insets). (b) Pancharatnam phase $P_{I \bar{E} R} (ε)$ for different $C$ and $ϕ$ . It detects the spin alignment. At $ϕ = 0^{+}$ , $P_{I \bar{E} R} (ε)$ covers different domains for different $C$ , regardless of junction details.Reuse & Permissions
Figure 3
Contribution of Pancharatnam phase to quantization. (a) A bound trajectory in a metallic box surrounded by insulators. Its quantization is affected by Pancharatnam phase from the geodesic polygon (see Bloch sphere) connecting the scattering states $(k_{j}, e_{j})$ involved along the trajectory. (b) Electron conductance $G (V_{g})$ along a topological waveguide $g$ with width $w$ , sandwiched between two insulators $j = l, r$ with band gap $Δ_{j = l, r}$ and Chern number $C_{j = l, r}$ ; $ε_{F}$ is inside the gaps $Δ_{j = l, r}$ . Inside the guide $g$ , gate voltage $V_{g}$ is applied and there is no gap ( $Δ_{g} = 0$ ). For $C_{l} = C_{r}$ ( $C_{l} \neq C_{r}$ ), conductance jumps by $e^{2} / h$ appear only within the shaded (white) domains of $V_{g}$ . The examples of $G (V_{g})$ are drawn for the cases of $C_{l} = C_{r}$ (blue solid line) and $C_{l} \neq C_{r}$ (red dashed).Reuse & Permissions
Figure 4
Geometric-phase device. (a) Topological Fabry-Pérot resonator, consisting of three regions $j = l, g, r$ with gap $Δ_{j = l, r}$ and gate voltage $V_{j = l, g, r}$ . (b) Transmission probability $τ (ϕ)$ of a plane wave, incoming from the left region $l$ (see left panel) or right region $r$ (right panel) with incidence angle $ϕ$ , through the resonator with $Δ_{l} = - Δ_{r}$ and $V_{l} = V_{r}$ . The results for different $V_{g}$ are drawn by curves of different style. (c) Multiterminal field-effect transistor based on the resonator in a Y junction; region $r$ now has two drains, D1 and D2, which are mirror-reflection symmetric. In the right panel, $Δ G (V_{g}) \equiv (G_{1} (V_{g}) - G_{2} (V_{g})) / (G_{1} (V_{g}) + G_{2} (V_{g}))$ is drawn as a function of $V_{g}$ at zero temperature in the zero-bias limit, where $G_{1}$ ( $G_{2}$ ) is electron conductance from source S to D1 (D2). While $G_{1} = G_{2}$ is always satisfied for $Δ_{1} = Δ_{2}$ , $Δ G$ is generally nonzero for $Δ_{l} = - Δ_{r}$ , although the setup has the mirror reflection symmetry. For the details of this figure, see Appendix .Reuse & Permissions

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