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Spontaneous edge current in higher chirality superconductors

Xin Wang, Zhiqiang Wang, and Catherine Kallin
Phys. Rev. B 98, 094501 – Published 4 September 2018
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  • INTRODUCTION
  • FORMALISM
  • EDGE CURRENTS WITHOUT MEISSNER SCREENING…
  • MEISSNER SCREENING EFFECT ON THE EDGE…
  • ROUGH SURFACE EFFECT
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The effects of finite temperature, Meissner screening, and surface roughness on the spontaneous edge current for higher chirality quasi-two-dimensional superconductors are studied in the continuum limit using the quasiclassical Eilenberger equations. We find that the total spontaneous current is nonzero at finite temperature T and maximized near T=Tc/2, where Tc is the transition temperature, although it vanishes at T=0. In the presence of surface roughness, we observe a surface current inversion in the chiral d-wave case that can be understood in terms of a disorder-induced s-wave pairing component in the rough surface regime. This conclusion is supported by a Ginzburg-Landau analysis. However, this current inversion is nonuniversal beyond the continuum limit, as demonstrated by self-consistent lattice Bogoliubov-de Gennes calculations.

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    • Received 26 May 2018

    DOI:https://doi.org/10.1103/PhysRevB.98.094501

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Impurities in superconductorsMeissner effectSuperconductivityTopological superconductors
    1. Techniques
    Bogoliubov-de Gennes equationsLandau-Ginzburg theory
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Xin Wang1, Zhiqiang Wang1, and Catherine Kallin1,2

    • 1Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada
    • 2Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

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    Issue

    Vol. 98, Iss. 9 — 1 September 2018

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    Images

    • Figure 1
      Figure 1

      Temperature dependence of the integrated edge current, Iy(T), for chiral d- (red open circles) and f- (green triangles) waves with the self-consistently determined superconducting order parameter. The black dots (open circles) are numerical results for chiral d (f)-wave with a uniform order parameter Δ1(x)=Δ2(x)≡Δ(bulk). Iy is scaled by J0ξ0, where J0=evFNFTc and ξ0=vF/πΔ(bulk) is the zero temperature coherence length.

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    • Figure 2
      Figure 2

      Edge state energy dispersion for chiral d-wave pairing obtained from BdG calculations with a self-consistently determined superconducting order parameter (solid red lines) and a uniform order parameter (dashed black lines). The shaded regimes represent dense bulk energy spectra, whose details are not shown here. Inside the bulk superconducting gap, there are two edge state energy dispersions crossing E=0 at ky=±kF/2.

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    • Figure 3
      Figure 3

      (a)–(c) Spatial dependencies of the edge current density, Jy(x), induced magnetic field, Bz(x), and vector potential, Ay(x), with Meissner screening taken into account for chiral p-, d- and f-wave pairings, respectively. GL ratio κ≡λL/ξ0=2.5. Jy(x), Ay(x) and Bz(x) are scaled by J0=evFNFTc, Δ(bulk)/evF and Bc=Φ0/22πξ0λL, respectively, where Φ0=h/2e and T=0.02Tc.

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    • Figure 4
      Figure 4

      (a)–(c) Spatial dependence of Δ1 and Im(Δ2) in the presence of surface roughness for chiral p-, d- and f-waves, respectively. (d)–(f) Spatial dependence of the edge current Jy(x) and the induced Bz(x) in the presence of surface roughness for different chiral pairing channels. The effective rough regime with width W=5ξ0 is shaded in grey. The strength of the roughness is characterized by the shortest mean free path, ℓp≡vFτ(x=0), in the rough regime, which is ξ0/ℓp=1.0 for the results shown. The two order parameter components have been already scaled by their bulk values. Meissner screening is not taken into account. T=0.02Tc.

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    • Figure 5
      Figure 5

      Comparison of Jy(x) obtained with (solid black line) and without (dashed red line) the off-diagonal impurity self energy, Σo.d., discussed in the text. The current from the GL free-energy analysis in Eq. (18) (blue open circles) is plotted. Green line with dots shows that the inverted current is greatly reduced when an s-wave repulsive interaction Vs=−5Vd is present. Here Vd is the bulk d-wave attractive interaction.

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    • Figure 6
      Figure 6

      Spatial dependencies of Jy(x), Bz(x), and Ay(x) with Meissner screening for chiral p-, d- and f-wave pairing (from top to bottom), in the presence of a rough surface in a region of width W=5ξ0 and with ℓp=ξ0.

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    • Figure 7
      Figure 7

      (a)–(c) Spatial dependence of Δ1 and Im(Δ2) with specular surface for chiral p-, d- and f-waves, respectively. (d)–(f) Spatial dependence of the edge current Jy(x) and the induced Bz(x) with specular surface for different chiral pairing channels. Note the different vertical scales in (d)–(f). Meissner screening is not taken into account. T=0.02Tc.

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    • Figure 8
      Figure 8

      Decomposition of the current density for the chiral d-wave pairing into different branches (i.e., different ky ranges). Inset: edge state energy dispersion for a chiral d-wave superconductor obtained from a lattice BdG calculation; the horizontal axis is ky/π; the two blue solid lines are edge state dispersions while the grey shaded regime represent the bulk state energy spectrum.

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    • Figure 9
      Figure 9

      Edge currents for the chiral d-wave pairing obtained from self-consistent BdG on a triangular lattice with different edge directions and chemical potentials. The shaded regime on the left has surface roughness, while the right surface is specular without disorder. We choose the rough regime width to be W=5 lattice sites. A relatively larger temperature T=0.1Tc has been chosen to reduce the Friedel oscillations in the current. The impurity potential strength, Vimp=15t, and the impurity density, nimp=0.2 per site, are large such that the effective local mean free path is short, same as in the Eilenberger calculation, where current reversal is seen.

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