Chiral zero modes in superconducting nanowires with Dresselhaus spin-orbit coupling

Hsien-chung Kao

Phys. Rev. B 90, 245435 – Published 29 December 2014

Abstract

Using chiral decomposition, we are able to find analytically the zero modes and the conditions for such modes to exist in the Kitaev ladder model and superconducting nanowires with Dresselhaus spin-orbit coupling. As a result, we are able to calculate the number of zero modes in these systems for arbitrary given parameters in the semi-infinite limit. Moreover, we find that when suitable resonance condition is satisfied exact zero modes exist even in finite systems. Because of this, the energy of the lowest positive energy shows oscillation with respect to the chemical potential.

Received 17 November 2014
Revised 17 December 2014

DOI:https://doi.org/10.1103/PhysRevB.90.245435

Authors & Affiliations

Hsien-chung Kao^*

Physics Department, National Taiwan Normal University, Taipei 11677, Taiwan and Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada

^*hckao@ntnu.edu.tw

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Issue

Vol. 90, Iss. 24 — 15 December 2014

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Figure 1
$E_{0}$ vs chemical potential $μ$ for the Kitaev chain is shown, where $E_{0}$ is the energy of the lowest positive energy mode. We choose $N = 10$ so that the energy difference between the exact zero modes and would-be zero modes is more manifest. Top: When $w > | Δ |$ (numerically $Δ = 0.6$ here), there are a pair of exact zero modes when the condition in Eq. (12) is satisfied. The value of the exact zero mode energy is zero up to numerical inaccuracy, which is of the order $10^{- 16} .$ Bottom: When $w < | Δ |$ , there is no zero mode even if $| μ | < 2 w .$ Here, we choose $Δ = 2.4$ so that the effect is more pronounced.
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Figure 2
$E_{0}, E_{1}$ vs chemical potential $μ$ for the Kitaev ladder models with $N_{x} = N_{y} = 6$ is shown, where $E_{0}, E_{1}$ are the energies of the two lowest positive energy modes. Top: For $w > | Δ |$ (numerically $Δ = 0.8$ here), there will be exact zero modes when the condition in Eq. (23) is satisfied. The two numbers in the parenthesis are the corresponding $k_{x}$ and $k_{y}$ for the mode, and there are more than one pair of exact zero modes in some specific values of $μ$ as predicted in Eq. (24) and (25). Again, the value of the exact zero mode energy is zero up to numerical inaccuracy, which is of the order $10^{- 16} .$ Bottom: For $w < | Δ |$ , there is no zero mode even if $| μ | < 4 w .$ Here, we choose $Δ = 2.4$ so that the effect is more pronounced.
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Figure 3
$E_{0, k_{y}}$ vs chemical potential $μ$ for Kitaev ladder models with $N_{x} = 120, N_{y} = 6$ is shown, where $E_{0, k_{y}}$ are the energies of the six lowest positive energy modes. $w_{x} = w_{y} = w$ is used as the unit of the energy. Top: For $w > | Δ |$ (numerically $Δ = 0.8$ here), a pair of zero modes with a specific $k_{y}$ will exist when the corresponding condition in Eq. (28) is satisfied. The dashed lines indicates the values of $μ$ , where various zero modes of the semi-infinite system begin to develop a gap. The numerical values are in good agreement with the analytic ones, and the discrepancy arises from the finiteness of $N_{x}$ used in the numerical calculation. Bottom: For $w < | Δ |$ , there is no zero mode even if $| μ | < 4 w .$ Here, we choose $Δ = 6.0$ so that the effect is more pronounced.
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Figure 4
$E_{0, k_{y}}$ vs chemical potential $μ$ for nanowires with Dresselhaus spin-orbit coupling is shown with $N_{x} = 120, N_{y} = 6$ . Here, $E_{0, k_{y}}$ are the energies of the six lowest positive energy modes, with $k_{y} = 1, \dots, 6$ from left to right. The dotted and dashed lines indicates the lower and upper critical values of $μ$ respectively, beyond which various zero modes of the semi-infinite system begin to develop a gap. Top: For $| V_{0} | > | Δ_{0} |$ and $2 w > \sqrt{V_{0}^{2} - Δ_{0}^{2}}$ , (numerically $V_{0} = 0.8, Δ_{0} = 0.64$ ), a pair of zero modes with a specific $k_{y}$ will exist when the corresponding condition in Eq. (35) is satisfied. Bottom: For $| V_{0} | > | Δ_{0} |$ and $2 w < \sqrt{V_{0}^{2} - Δ_{0}^{2}}$ (numerically $V_{0} = 3.6, Δ_{0} = 0.4$ ), a pair of zero modes with a specific $k_{y}$ will exist when the corresponding condition in Eq. (36) is satisfied.
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Figure 5
$E_{0, k_{y}}$ vs chemical potential $μ$ for nanowires with Dresselhaus spin-orbit coupling with $N_{x} = 120, N_{y} = 6$ is shown. Here, $E_{0, k_{y}}$ are the energies of the six lowest positive energy modes, with $k_{y} = 1, \dots, 6$ from left to right. For $| V_{0} | < | Δ_{0} |$ , there is no zero mode for any value of $μ .$ Numerically, we choose $V_{0} = 0.6, Δ_{0} = 1.3$ here.
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Figure 6
$E_{0}$ vs chemical potential $μ_{eff}$ for nanowires with Dresselhaus spin-orbit coupling is shown, where $E_{0}$ is the energy of the lowest positive energy mode. Here, we choose $λ_{D} = 0.5$ , $V_{0} = 0.7$ , and $N_{x} = 20$ Top: For $| V_{0} | > | Δ_{0} |$ (numerically, $Δ_{0} = 0.1$ ), there are a pair of exact zero modes when the conditions in Eqs. (A1) and (A2) are satisfied. The value of the exact zero mode energy is zero up to numerical inaccuracy, which is of the order $10^{- 16} .$ Bottom: For comparison, when $| V_{0} | < | Δ_{0} |$ (numerically $Δ_{0} = 0.8$ ), there are no zero modes.
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