Chiral nodes and oscillations in the Josephson current in Weyl semimetals

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Chiral nodes and oscillations in the Josephson current in Weyl semimetals

Udit Khanna, Dibya Kanti Mukherjee, Arijit Kundu, and Sumathi Rao

Phys. Rev. B 93, 121409(R) – Published 23 March 2016

Abstract

The separation of the Weyl nodes in a broken time-reversal symmetric Weyl semimetal leads to helical quasiparticle excitations at the Weyl nodes which, when coupled with overall spin conservation, allows only internodal transport at the junction of the Weyl semimetal with a superconductor. This leads to an unusual periodic oscillation in the Josephson current as a function of $k_{0} L$ , where $L$ is the length of the Weyl semimetal and $2 k_{0}$ is the internodal distance. This oscillation is robust and should be experimentally measurable, providing a direct path to confirming the existence of chiral nodes in the Weyl semimetal.

Received 15 September 2015

DOI:https://doi.org/10.1103/PhysRevB.93.121409

Physics Subject Headings (PhySH)

Josephson effect Weyl fermions

Semimetals

Chiral symmetry T-symmetry

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Udit Khanna¹, Dibya Kanti Mukherjee¹, Arijit Kundu^2,3, and Sumathi Rao¹

¹Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
²Physics Department, Technion, 320003 Haifa, Israel
³Department of Physics, Indiana University, Bloomington, Indiana 47405, USA

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Issue

Vol. 93, Iss. 12 — 15 March 2016

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Images

Figure 1
(a) The setup for Josephson current with two superconductors (SC) characterized by phases $ϕ_{L}$ and $ϕ_{R}$ sandwiching a WSM of length $L$ between them where a time-reversal broken perturbation separates the Weyl nodes in momentum space by $2 k_{0}$ in $k_{z}$ . (b) The Josephson current (at normal incidence) is periodic in $L$ with a period of $π / k_{0}$ . We show the zero-temperature Josephson current, Eq. (9), as a function of the superconducting phase difference $ϕ$ , for various values of $L$ in solid (dashed) lines for $θ$ between 0 and $π$ ( $π$ and $2 π$ ), where $θ = 2 k_{0} L mod (2 π)$ . The parameters used are $k_{0} L = 31.4, 32.0, 32.5, 33, 33.5, 34.0, 34.454, ℏ^{2} k_{0}^{2} / 2 m_{W} = 10 μ_{W} = 10^{3} Δ = μ_{S} / 2, p ≪ k_{0}$ , and $m_{S} = m_{W}$ .
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Figure 2
(a) Both reflection (R) and Andreev reflection (AR) in WSM occur from one chiral node to another. The chiralities of the nodes are denoted as $+ ve$ and $- ve$ , whereas the solid and the dashed lines show dispersions of Eq. (2) with positive and negative velocities ( $= d E / d k$ ). The bands have opposite spins, which accounts for the change of chirality for both normal and Andreev reflection. (b) The probability that an electron will be reflected as an electron ( $| χ^{\pm} |^{2}$ ) or as a hole ( $| η^{\pm} |^{2}$ ) at a WSM-SC interface, discussed in Eq. (5). Note that the probability of reflection as an electron is finite. The parameters used are the same as in Fig. 1.
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Figure 3
The variation of the bound levels [solutions of Eq. (6)] near the chemical potential with the length $L$ of the WSM for various values of $θ$ , where $θ = 2 k_{0} L mod (2 π)$ . The parameters used are the same as in Fig. 1.
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Figure 4
The Josephson current as a function of both $L$ and $k_{0}$ is shown at the value of $ϕ \approx π / 2$ . The initial value at the origin is $(k_{0}, L) = (q, l), q l \approx 10 π$ . The contours of constant current follow a set of (approximate) hyperbolas for constant $θ = 2 k_{0} L mod (2 π)$ , a few of which are shown in the right margin (with the minimum and the maximum current occurring near $θ = π$ and $θ = 0$ , respectively). Other parameters used are the same as in Fig. 1.
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Figure 5
Result of lattice simulation: (a) Josephson current as a function of the superconducting phase difference $ϕ$ for various values of $θ = 2 k_{0} L mod (2 π)$ . (b) Josephson current as a function of the length ( $k_{0} L / π$ ) at $ϕ = π / 2$ . The parameters used are [42] $\tilde{t} = 0.25 t$ [see Eq. (10)], $ε = 6 t, λ = λ_{z} = t, Δ = 0.01 t, μ = 0.02 t$ , and $L = 60$ sites. For these parameters, the position of Weyl nodes, $k_{0} = b_{z} / λ_{z}$ (the lattice constant is taken as the unit of length).
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