# Relation of Superconducting Pairing Symmetry and Non-Magnetic Impurity Effects in Vortex States

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## Abstract

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## 1. Introduction

## 2. Formulation

#### 2.1. Eilenberger Theory with Non-Magnetic Impurity Scatterings

#### 2.2. Methods of Numerical Calculations in the Vortex Lattice

## 3. Impurity Effects in Uniform States

## 4. Impurity Effects in Vortex States for ${p}_{x}$-Wave Pairing ${\phi}_{{p}_{x}}(\theta )$ and Anisotropic s-Wave ${\phi}_{|{p}_{x}|}(\theta )$

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Some pairing functions $\phi (\theta )$ are schematically presented on a circular Fermi surface in $({k}_{x},{k}_{y})$ plane. (

**a**) ${\phi}_{{p}_{x}}(\theta )=\sqrt{2}cos\theta $ for the ${p}_{x}$-wave symmetry. (

**b**) ${\phi}_{|{p}_{x}|}(\theta )=|{\phi}_{{p}_{x}}(\theta )|=\sqrt{2}|cos\theta |$ for anisotropic s-wave symmetry which has the same amplitude as ${\phi}_{{p}_{x}}(\theta )$.

**Figure 2.**(

**a**) Amplitude of the pair potential $\mathsf{\Delta}$ as a function of $1/{\tau}_{0}$ in uniform states at a zero field. $\mathsf{\Delta}$ is plotted in the range $0\le 1/{\tau}_{0}\le 0.2$ for ${p}_{x}$-wave pairing $\phi (\theta )={\phi}_{{p}_{x}}(\theta )$ in the Born and the unitary limits, with $\mathsf{\Delta}$ for anisotropic s-wave paring $\phi (\theta )={\phi}_{|{p}_{x}|}(\theta )$ in the Born limit. $T/{T}_{\mathrm{c}}=0.5$. (

**b**) $\mathsf{\Delta}$ for anisotropic s-wave paring $\phi (\theta )={\phi}_{|{p}_{x}|}(\theta )$ is plotted in wider range $0\le 1/{\tau}_{0}\le 25$ in the Born limit. $\mathsf{\Delta}$ in the unitary limit shows almost the same $1/{\tau}_{0}$-dependence.

**Figure 3.**DOS spectrum $N(E)$ in uniform states at a zero field. (

**a**) $N(E)$ for the ${p}_{x}$-wave pairing $\phi (\theta )={\phi}_{{p}_{x}}(\theta )$. $1/{\tau}_{0}=0.01$, 0.05, 0.1, and 0.15 in the Born limit. (

**b**) The same as (

**a**), but in the unitary limit. (

**c**) $N(E)$ for anisotropic s-wave paring $\phi (\theta )={\phi}_{|{p}_{x}|}(\theta )$. $1/{\tau}_{0}=0.01$, 0.1, 0.6, and 1.0 in the Born limit. (

**d**) The same as (

**c**), but in the unitary limit.

**Figure 4.**(

**a**) Spatial structure of the pair potential $|\mathsf{\Delta}(\mathbf{r})|$ around a vortex core within a unit cell of the vortex lattice in $xy$ plane for the ${p}_{x}$-wave pairing $\phi (\theta )={\phi}_{{p}_{x}}(\theta )$. $1/{\tau}_{0}=0.01$, 0.05, 0.10, and 0.15. The upper (middle) panels are for the Born (unitary) limit. Lower panels show profile of $|\mathsf{\Delta}(\mathbf{r})|$ along the x (red lines) and the y axes (blue lines). The solid (dashed) lines are for the Born (unitary) limit. (

**b**) The same as panel (a), but for anisotropic s wave pairing symmetry $\phi (\theta )={\phi}_{|{p}_{x}|}(\theta )$. $1/{\tau}_{0}=0.01$, 0.1, 0.6, and 1.0.

**Figure 5.**(

**a**) Impurity scattering rate $1/{\tau}_{0}$ dependence of vortex core radius ${\xi}_{x}$ and ${\xi}_{y}$, which are determined from the slope of $|\mathsf{\Delta}(\mathbf{r})|$ at the vortex center, for ${\phi}_{{p}_{x}}(\theta )$ in the range $0\le 1/{\tau}_{0}\le 0.15$ and ${\phi}_{|{p}_{x}|}(\theta )$ in the range $0\le 1/{\tau}_{0}\le 1$. The Born (solid lines) and the unitary (dashed lines) limit shows almost identical $1/{\tau}_{0}$ dependence. (

**b**) The anisotropy ratio ${\xi}_{y}/{\xi}_{x}$ as a function of $1/{\tau}_{0}$, from ${\xi}_{x}$ and ${\xi}_{y}$ in panel (

**a**).

**Figure 6.**(

**a**) Spatial structure of zero-energy LDOS $N(E=0,\mathbf{r})$ around a vortex core in $xy$ plane for the ${p}_{x}$-wave pairing $\phi (\theta )={\phi}_{{p}_{x}}(\theta )$. $1/{\tau}_{0}=0.01$, 0.05, 0.10, and 0.15. The upper (middle) panels are for the Born (unitary) limit. Lower panels show profile of $N(E=0,\mathbf{r})$ along the x (red lines) and the y axes (blue lines). The solid (dashed) lines are for the Born (unitary) limit. (

**b**) The same as panel (a), but for anisotropic s wave pairing symmetry $\phi (\theta )={\phi}_{|{p}_{x}|}(\theta )$. $1/{\tau}_{0}=0.01$, 0.1, 0.6, and 1.0.

**Figure 7.**(

**a**) Peak height $N(E=0,r=0)$ at the vortex center as a function of impurity scattering rate $1/{\tau}_{0}$ for ${\phi}_{{p}_{x}}(\theta )$ in the range $0\le 1/{\tau}_{0}\le 0.15$ and ${\phi}_{|{p}_{x}|}(\theta )$ in the range $0\le 1/{\tau}_{0}\le 1$. The solid (dashed) lines are for the Born (unitary) limit. (

**b**) $1/{\tau}_{0}$ dependence of vortex core radius ${\xi}_{x}$ and ${\xi}_{y}$, which are determined from the zero energy LDOS $N(E=0,\mathbf{r})$ around a vortex core as described in text, for ${\phi}_{{p}_{x}}(\theta )$ and ${\phi}_{|{p}_{x}|}(\theta )$. Solid (dashed) lines are for the Born (unitary) limit. (

**c**) The anisotropy ratio ${\xi}_{y}/{\xi}_{x}$ as a function of $1/{\tau}_{0}$, from ${\xi}_{x}$ and ${\xi}_{y}$ in panel (b).

**Figure 8.**(

**a**) Spatial structure of LDOS $N(E,\mathbf{r})$ around a vortex core in $xy$ plane for the ${p}_{x}$-wave pairing $\phi (\theta )={\phi}_{{p}_{x}}(\theta )$ at $E=0.01$, 0.1, 0.2, and 0.5. $1/{\tau}_{0}=0.01$ (upper panels), 0.10 (middle panels) in the Born limit, and $1/{\tau}_{0}=0.1$ in the unitary limit (lower panels). (

**b**) Profile of $N(E,\mathbf{r})$ along the x (red lines) and the y axes (blue lines) for the ${p}_{x}$-wave pairing $\phi (\theta )={\phi}_{{p}_{x}}(\theta )$. The solid (dashed) lines are for the Born (unitary) limit. $1/{\tau}_{0}=0.01$ (upper panels) and 0.1 (lower panels).

**Figure 9.**The same as Figure 8, but for anisotropic s wave pairing symmetry ${\phi}_{|{p}_{x}|}(\theta )$.

**Figure 10.**(

**a**) Bold line schematically shows main distribution of the $\mathbf{k}$-resolved LDOS $N(E,\mathbf{k},\mathbf{r})$. Vortex center is located at O(0,0). See text for details. (

**b**) Quasiparticle trajectories at low E for the ${p}_{x}$-wave pairing symmetry ${\phi}_{{p}_{x}}(\theta )$ are schematically presented. (

**c**) The same as panel (

**b**), but at higher E.

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**MDPI and ACS Style**

Sera, Y.; Ueda, T.; Adachi, H.; Ichioka, M.
Relation of Superconducting Pairing Symmetry and Non-Magnetic Impurity Effects in Vortex States. *Symmetry* **2020**, *12*, 175.
https://doi.org/10.3390/sym12010175

**AMA Style**

Sera Y, Ueda T, Adachi H, Ichioka M.
Relation of Superconducting Pairing Symmetry and Non-Magnetic Impurity Effects in Vortex States. *Symmetry*. 2020; 12(1):175.
https://doi.org/10.3390/sym12010175

**Chicago/Turabian Style**

Sera, Yasuaki, Takahiro Ueda, Hiroto Adachi, and Masanori Ichioka.
2020. "Relation of Superconducting Pairing Symmetry and Non-Magnetic Impurity Effects in Vortex States" *Symmetry* 12, no. 1: 175.
https://doi.org/10.3390/sym12010175