Strain-engineered Majorana zero energy modes and ${\ensuremath{\varphi}}_{0}$ Josephson state in black phosphorus

Strain-engineered Majorana zero energy modes and $φ_{0}$ Josephson state in black phosphorus

Mohammad Alidoust, Morten Willatzen, and Antti-Pekka Jauho

Phys. Rev. B 98, 085414 – Published 7 August 2018

Abstract

We develop a theory for strain control of Majorana zero energy modes and the Josephson effect in black phosphorus (BP) devices proximity coupled to a superconductor. Employing realistic values for the band parameters subject to strain, we show that the strain closes the intrinsic band gap of BP; however, the proximity effect from the superconductor reopens it and creates Dirac and Weyl nodes. Our results illustrate that Majorana zero energy flat bands connect the nodes within the band-inverted regime in which their associated density of states is localized at the edges of the device. In a ferromagnetically mediated Josephson configuration, the exchange field induces superharmonics in the supercurrent phase relation in addition to a $φ_{0}$ phase shift, corresponding to a spontaneous supercurrent, and strain offers an efficient tool to control these phenomena. We analyze the experimental implications of our findings and show that they can pave the way for creating a rich platform for studying two-dimensional Dirac and Weyl superconductivity.

Received 14 March 2018

DOI:https://doi.org/10.1103/PhysRevB.98.085414

Physics Subject Headings (PhySH)

Majorana fermions

2-dimensional systems Phosphorene Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Mohammad Alidoust¹, Morten Willatzen^2,3, and Antti-Pekka Jauho⁴

¹Department of Physics, K. N. Toosi University of Technology, Tehran 15875-4416, Iran
²Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, No. 30 Xueyuan Road, Haidian District, Beijing 100083, China
³Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark
⁴Center for Nanostructured Graphene, Department of Micro- and Nanotechnology, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

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Vol. 98, Iss. 8 — 15 August 2018

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Figure 1
Schematic of the proposed devices made of a phosphorene monolayer. The plane of black phosphorus is located in the $x y$ plane, and strain is exerted into the plane of the setups in the $y$ direction. (a) A single $s$ -wave superconducting electrode (S) of length $d_{S}$ is proximity coupled to black phosphorus. In order to reveal the density of states in the vicinity of the junction location along the $x$ axis, the STM tip should be placed close to the superconductor edge and black phosphorus. (b) A ferromagnetic (F) phosphorene Josephson junction of length $d_{F}$ and width $W_{F}$ .
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Figure 2
Band structure of a phosphorene monolayer, finite sized in the $y$ direction, as a function of momentum in the $x$ direction, i.e., $k_{x}$ . (a) Strains $ɛ_{x x}, ɛ_{y y}$ are set to zero, and superconducting gap $| Δ | \neq 0$ . (b) $ɛ_{x x} = - 0.2, ɛ_{y y} = 0$ , and $| Δ | = 0$ . (c) $ɛ_{x x} = - 0.2, ɛ_{y y} = 0$ , and $| Δ | \neq 0$ . (d) $ɛ_{x x} = 0, ɛ_{y y} = - 0.2$ , and $| Δ | \neq 0$ .
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Figure 3
Supercurrent density $J_{y} (φ)$ as a function of superconducting phase difference $φ$ . (a) Different values of magnetization strength are considered: $h_{z} = 0, 0.05, 0.1, 0.15, 0.2$ , and 0.25 eV, where we have set $ɛ_{x x} = - 0.25$ and $ɛ_{y y} = 0$ . (b) Magnetization strength is set to $h_{z} = 0.1$ eV, and the current density is examined for differing values of $ɛ_{x x} = 0, - 0.05, - 0.1, - 0.125, - 0.15, 0.175, - 0.2$ , and $- 0.25$ where $ɛ_{y y} = 0$ .
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Figure 4
(a)–(d) Band structure of finite-size black phosphorus in the $y$ direction as a function of momentum in the $x$ direction, i.e., $k_{x}$ . The parameters are set equal to those of Fig. 2, except now the chemical potential possesses a nonzero value, $μ = 1.5$ eV. (a) $ɛ_{x x} = - 0.25$ and $ɛ_{y y} = - 0.25$ . (b) $ɛ_{x x} = + 0.25$ and $ɛ_{y} = - 0.25$ . (c) $ɛ_{x x} = - 0.25$ and $ɛ_{y y} = + 0.25$ . (d) $ɛ_{x x} = + 0.25$ and $ɛ_{y y} = + 0.25$ . (e1) and (e2) Band structure of superconductive BP with parameters set identical to those in (a), except now no boundaries are applied to the system. (f1) and (f2) Band structure of infinite superconductive BP without any boundaries with parameters set identical to those in (d). In (g), (h1), and (h2) we set $| Δ | \neq 0, d_{S} = 40$ nm, $ɛ_{x x} = - 0.2$ , and $ɛ_{y y} = k_{x} = 0$ . (g) Eigenenergies $E_{ℓ}$ vs the energy index $ℓ$ . (h1) and (h2): The absolute wave function $| ψ (y) |$ vs the location $y (nm) \in [0, d_{S}]$ inside the system, corresponding to different eigenenergies presented in (g).
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Figure 5
Supercurrent density $J_{y} (φ)$ in the $y$ direction for various values of magnetization strength $h_{z} = 0, 0.05, 0.1, 0.15, 0.2$ , and 0.25 eV, where we have fixed the chemical potential at $μ = 1.5$ eV. (a) $ɛ_{x x} = - 0.25$ and $ɛ_{y y} = - 0.25$ . (b) $ɛ_{x x} = + 0.25$ and $ɛ_{y y} = - 0.25$ . (c) $ɛ_{x x} = - 0.25$ and $ɛ_{y y} = + 0.25$ . (d) $ɛ_{x x} = + 0.25$ and $ɛ_{y y} = + 0.25$ .
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Physical Review B

covering condensed matter and materials physics

Strain-engineered Majorana zero energy modes and $φ_{0}$ Josephson state in black phosphorus

Mohammad Alidoust, Morten Willatzen, and Antti-Pekka Jauho

Phys. Rev. B 98, 085414 – Published 7 August 2018

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Physical Review B

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Strain-engineered Majorana zero energy modes and φ0 Josephson state in black phosphorus

Mohammad Alidoust, Morten Willatzen, and Antti-Pekka Jauho

Phys. Rev. B 98, 085414 – Published 7 August 2018

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Strain-engineered Majorana zero energy modes and $φ_{0}$ Josephson state in black phosphorus