# London Penetration Depth as a Test of Order Parameter Symmetry in Sodium Cobaltate Superconductors

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## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Equation for the London Penetration Depth

**A**. A comprehensive description of the methods for calculating the London penetration depth can be found in the textbooks [31,32] and original papers [34,35,36,37]. We shall follow the paper [37] where a simple and physically clear derivation of the formula for $\lambda $ is given for arbitrary dispersion laws and gap functions of any symmetry. The formula reads as follows

## 4. Temperature Dependence of the London Penetration Depth for the Chiral Order Parameter

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HTSC | High Temperature Super Conductor |

ARPES | Angle-Resolved Photo Emission Spectroscopy |

LDA | Local Density Approximation |

DMFT | Dynamical Mean-Field Theory |

FRG | Functional Renormalization Group |

## Appendix A

## Appendix B

## References

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**Figure 1.**The Brillouin zone of triangular lattice of cobalt ions. The red solid line represents the Fermi contour at $x=0.29$. The blue dash-dotted and green dashed lines show zeros position for respectively real and imaginary parts of the chiral order parameter ${\mathrm{\Delta}}_{k}$. The points of the Brillouin zone where all three lines intersect (marked by green circles) indicate the position of the Dirac points of the Bogolyubov spectrum ${E}_{k}$. The model parameters are: $t=0.123$ eV, ${\mathrm{\Delta}}_{1}=0.23\xb7{\mathrm{\Delta}}_{2}$, ${\mathrm{\Delta}}_{2}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1.76\phantom{\rule{3.33333pt}{0ex}}{T}_{c}$, ${T}_{c}$ = 5 K.

**Figure 2.**Temperature dependencies of ${\lambda}^{-2}$ in Na${}_{x}$CoO${}_{2}\xb7y$H${}_{2}$O with the chiral (d+id) order parameter for different values of x: the solid curve is for $x=0.291$; dashed — $x=0.253$; dash-dotted — $x=0.331$. Parameters of the model are: $t=0.123$ eV, ${\mathrm{\Delta}}_{1}=0.23\xb7{\mathrm{\Delta}}_{2}$, ${\mathrm{\Delta}}_{2}=1.76\phantom{\rule{3.33333pt}{0ex}}{T}_{c}$, ${T}_{c}$ = 5 K. The insert shows the same curves on the enlarged scale.

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**MDPI and ACS Style**

Dzebisashvili, D.M.; Klyuchantsev, A.B.
London Penetration Depth as a Test of Order Parameter Symmetry in Sodium Cobaltate Superconductors. *Symmetry* **2019**, *11*, 633.
https://doi.org/10.3390/sym11050633

**AMA Style**

Dzebisashvili DM, Klyuchantsev AB.
London Penetration Depth as a Test of Order Parameter Symmetry in Sodium Cobaltate Superconductors. *Symmetry*. 2019; 11(5):633.
https://doi.org/10.3390/sym11050633

**Chicago/Turabian Style**

Dzebisashvili, Dmitry M., and Andrey B. Klyuchantsev.
2019. "London Penetration Depth as a Test of Order Parameter Symmetry in Sodium Cobaltate Superconductors" *Symmetry* 11, no. 5: 633.
https://doi.org/10.3390/sym11050633