**Appendix**

In this appendix, the coefficients in Eqs. 13-18 aregiven. For d-wave superconducting regions, they are defined as,

$$P\_{\vec{y}}^{dd}\left(\{\mathbf{A}\}\right) \equiv \frac{3}{4} \mathcal{J}\_d^2 \left(\Delta\_d^0\right)^2 \left(\sum\_{a=x,y} K\_{\vec{y}}^{aa} + \sum\_{\substack{i,j\\a=x,y}} I\_{i\vec{y},j\vec{y}} A\_{i\vec{a}} A\_{i\vec{a}}\right) - \frac{3}{4} \left(\Delta\_d^0\right)^2 I\_{\vec{y}}\tag{25}$$

$$\begin{aligned} P\_{\boldsymbol{y}}^{d\ell\Delta t} \left( \{\boldsymbol{\Delta}\} \right) & \equiv \left( \frac{\boldsymbol{\Delta}\_{\boldsymbol{s}}}{\boldsymbol{\Delta}\_{\boldsymbol{d}}} \right)^{2} \sum\_{i\_{\boldsymbol{i}} \scriptstyle{j}} I\_{i\_{\boldsymbol{i}} \scriptstyle{j} j} \left( \mathbf{3} \, \text{Re} \, \Delta\_{\boldsymbol{s} i\_{\boldsymbol{i}}} \, \text{Re} \, \Delta\_{\boldsymbol{s} i\_{\boldsymbol{i}}} + \text{Im} \, \Delta\_{\boldsymbol{s} i\_{\boldsymbol{i}}} \, \text{Im} \, \Delta\_{\boldsymbol{s} i\_{\boldsymbol{i}}} \right) \\ & + \frac{3}{4} \sum\_{i\_{\boldsymbol{i}} \scriptstyle{j}} I\_{i\_{\boldsymbol{i}} \scriptstyle{j} j} \left( \mathbf{3} \, \text{Re} \, \Delta\_{\boldsymbol{d} i\_{\boldsymbol{i}}} \, \text{Re} \, \Delta\_{\boldsymbol{d} i\_{\boldsymbol{i}}} + \text{Im} \, \Delta\_{\boldsymbol{d} i\_{\boldsymbol{i}}} \, \text{Im} \, \Delta\_{\boldsymbol{d} i\_{\boldsymbol{i}}} \right) \end{aligned} \tag{26}$$

$$\begin{aligned} P\_{ij}^{d\ell \ge 1} \left( \{ \Delta \} \right) & \equiv \left( \frac{\lambda\_s}{\lambda\_d} \right)^2 \sum\_{i\_l} I\_{i\_l \ge \mathcal{Y}} \left( \operatorname{Re} \Delta\_{s i\_l} \operatorname{Re} \Delta\_{s i\_2} + 3 \operatorname{Im} \Delta\_{s i\_l} \operatorname{Im} \Delta\_{s i\_2} \right) \\ & + \frac{3}{4} \sum\_{i\_l} I\_{i\_l \ge \mathcal{Y}} \left( \operatorname{Re} \Delta\_{d i\_l} \operatorname{Re} \Delta\_{d i\_1} + 3 \operatorname{Im} \Delta\_{d i\_1} \operatorname{Im} \Delta\_{d i\_2} \right) \end{aligned} \tag{27}$$

$$\mathcal{Q}\_{\boldsymbol{\uprho}}^{\boldsymbol{\uprho}\boldsymbol{\uprho}}\left(\{\mathbf{A}\}\right) \equiv -\frac{3}{4} \mathcal{J}\_{d}^{\boldsymbol{\uprho}} \left(\Delta\_{d}^{\boldsymbol{\uprho}}\right)^{2} \sum\_{\boldsymbol{\uprho}\_{\boldsymbol{i}\_{l}}} \left(J\_{\boldsymbol{\uprho}\_{\boldsymbol{i}\_{l}}j}^{\boldsymbol{\upalpha}} - J\_{\boldsymbol{\uprho}\_{\boldsymbol{i}\_{l}}i}^{\boldsymbol{\upalpha}}\right) A\_{\boldsymbol{i}\_{l}}^{\boldsymbol{\upalpha}}\tag{28}$$

$$\mathcal{Q}\_{ij}^{d\ell 2} \left( \{ \Delta \} \right) \equiv \left( \frac{\mathcal{\lambda}\_i}{\mathcal{\lambda}\_\ell} \right)^2 \sum\_{i \neq j} I\_{i\_1 j} \, \mathcal{Q} \operatorname{Im} \Delta\_{s i\_1} \operatorname{Re} \Delta\_{s i\_2} + \frac{3}{4} \sum\_{i \neq j} I\_{i\_1 j} \, \mathcal{Q} \operatorname{Im} \Delta\_{d\_1} \operatorname{Re} \Delta\_{d i\_2} \tag{29}$$

Composite Structures of d-Wave and s-Wave Superconductors (d-Dot): Analysis Using Two-Component Ginzburg-Landau Equations 339

$$P\_{\vec{y}}^{dv}\left(\{\mathbf{A}\}\right) \equiv \frac{3}{4} \mathcal{J}\_d^2 \left(\Delta\_d^0\right)^2 \frac{\mathcal{A}\_\circ}{\mathcal{A}\_d} \left(K\_{\vec{y}}^{xx} - K\_{\vec{y}}^{\prime\prime} + \sum\_{\substack{i\_1 \\ \alpha = x, y}} I\_{i\_1 i\_2 \vec{y}} \left(A\_{i\_1 \boldsymbol{\varpi}} A\_{i\_2 \boldsymbol{\uppi}} - A\_{i\_1 \boldsymbol{\uppi}} A\_{i\_2 \boldsymbol{\uppi}}\right)\right) \tag{30}$$

338 Superconductors – Materials, Properties and Applications

independent as shown in the insets of Fig. 19.

devices[27].

**Appendix** 

regions, they are defined as,

**5. Conclusions** 

free energy than that of the antiferromagnetic state. For longer distance, free energies of these states are almost same, because the spontaneous magnetic fluxes become almost

These interactions between d-dots can be used for changing the state of the d-dots [26]. Using these interactions, the d-dots have a potential applicability to the superconducting

We showed that the anisotropic pairing of superconductivity causesinteresting phenomena when two different superconductors are combined. Spontaneous half-quantized magnetic flux around the junction between d-wave and s-wave superconductors is one of such phenomena. It is simulated by the two-components Ginzburg-Landau equations, which can treat the anisotropy of d-wave superconductivity. This half-quantum magnetic state for the d-wave superconductors embedded in the s-wave superconductor has good properties for

In this appendix, the coefficients in Eqs. 13-18 aregiven. For d-wave superconducting

( ) { } ( ) ( ) 12 1 2 1 2 2 2 2 0 <sup>0</sup>

<sup>2</sup> 3Re Re Im Im

*ij i i ij si si si si*

*ij d d ij i i ij i i d ij x y ii*

3 3 4 4 αα

,

*x y*

α

( ) { } ( )

( ) { } ( )

Δ ≡ Δ Δ+ Δ Δ

( ) { } ( ) ( ) 1 11 1

( ) { } 1 2 <sup>1</sup> <sup>2</sup> 1 2 <sup>1</sup> <sup>2</sup> 1 2 1 2

Δ ≡ Δ Δ+ Δ Δ

<sup>2</sup> <sup>3</sup> 2Im Re 2Im Re

*ij i i ij si si i i ij di di d i i i i*

α

=

,

<sup>2</sup> 2 0

<sup>2</sup> Re Re 3Im Im

*ij i i ij si si si si*

Δ ≡ Δ Δ+ Δ Δ

=

 ≡Δ + − Δ

α α

(26)

(27)

*P* **A** *K I AA I* (25)

1 2 1 2 1 2

1 2 1 2 1 2

α

4

*QI I* (29)

 αα

*Q* **A** *J JA* (28)

,

( )

( )

1 2 1 2 1 2

*i i ij di di di di*

ξ

≡− Δ − *dd ij d d ii j ji i i i x y*

3 4

*dd <sup>s</sup>*

1 2 1 2 1 2

*i i ij di di di di*

α

1 2

1 2

<sup>3</sup> Re Re 3Im Im

+ Δ Δ+ Δ Δ

<sup>3</sup> 3Re Re Im Im

+ Δ Δ+ Δ Δ

2

*d i i*

*dd*

2

*d i i*

λ

λ

λ

λ

2

λ

λ

=

ξ

*dd R s*

*dd I s*

*P I*

*P I*

1 2

1 2

*i i*

*I*

4

*i i*

*I*

4

applications to superconducting devices, especially for classical bits or qubits.

$$\mathbb{E}\left[\underline{Q}\_{\boldsymbol{\vartheta}}^{\rm dis}\left(\{\mathbf{A}\}\right)\equiv\frac{3}{4}\mathcal{J}\_{d}^{2}\left(\Delta\_{d}^{0}\right)^{2}\frac{\hat{\mathcal{A}}\_{i}}{\hat{\mathcal{A}}\_{d}}\sum\_{i\_{\boldsymbol{i}}}\Big[\left(J\_{\boldsymbol{\vartheta}\_{\boldsymbol{i}}j}^{\boldsymbol{x}}-J\_{\boldsymbol{j}\boldsymbol{i}}^{\boldsymbol{x}}\right)A\_{i\_{\boldsymbol{i}}}^{\boldsymbol{x}}-\left(J\_{\boldsymbol{\vartheta}\_{\boldsymbol{i}}j}^{\boldsymbol{y}}-J\_{\boldsymbol{j}\boldsymbol{i}}^{\boldsymbol{y}}\right)A\_{i\_{\boldsymbol{i}}}^{\boldsymbol{y}}\Big]\tag{31}$$

$$W\_i^{d\mathbb{R}}\left(\{\Delta\}\right) \equiv \frac{3}{2} \sum\_{i\_l i\_l i\_l} I\_{i\_l i\_l i\_l} \Delta\_{i\_l} \Delta\_{i\_l}^\* \operatorname{Re} \Delta\_{i\_l} \tag{32}$$

$$W\_i^{dl} \left( \{ \Delta \} \right) \equiv \frac{3}{2} \sum\_{i \neq i\_l} I\_{i\_l i\_2 i\_2} \Delta\_{i\_l} \Delta\_{i\_2}^\* \operatorname{Im} \Delta\_{i\_3} \tag{33}$$

$$P\_{\vec{y}}^{us}\left(\left\{\mathbf{A}\right\}\right) \equiv \frac{3}{2} \mathcal{Z}\_d^2 \left(\Delta\_d^0\right)^2 \left(\frac{\mathcal{A}\_s}{\mathcal{A}\_d}\right)^2 \left(\sum\_{a=x,y} K\_{\vec{y}}^{aa} + \sum\_{\substack{i,j\\a\sim x,y}} I\_{i\ne j;i} A\_{i\cdot a} A\_{i\cdot a}\right) + 2 \left(\frac{\mathcal{A}\_s}{\mathcal{A}\_d}\right)^4 \left(\Delta\_s^0\right)^2 I\_{\vec{y}}\tag{34}$$

$$\begin{split} \mathbf{P}\_{\boldsymbol{\vartheta}}^{\boldsymbol{ss}\approx \boldsymbol{R}} \left( \{ \boldsymbol{\Lambda} \} \right) & \equiv \left( \frac{\boldsymbol{\mathcal{A}}\_{\boldsymbol{s}}}{\boldsymbol{\mathcal{A}}\_{\boldsymbol{d}}} \right)^{2} \sum\_{\boldsymbol{i}:\boldsymbol{i}\_{\boldsymbol{i}} \neq \boldsymbol{0}} \left( \mathbf{3} \, \text{Re}\,\boldsymbol{\Delta}\_{\boldsymbol{d}\_{\text{i}}} \, \text{Re}\,\boldsymbol{\Delta}\_{\boldsymbol{d}\_{\text{i}}} + \text{Im}\,\boldsymbol{\Delta}\_{\boldsymbol{d}\_{\text{i}}} \, \text{Im}\,\boldsymbol{\Delta}\_{\boldsymbol{d}\_{\text{i}}} \right) \\ + 2 \left( \frac{\boldsymbol{\mathcal{A}}\_{\boldsymbol{s}}}{\boldsymbol{\mathcal{A}}\_{\boldsymbol{d}}} \right)^{4} \sum\_{\boldsymbol{i}:\boldsymbol{i}} I\_{\boldsymbol{i}\boldsymbol{i}\boldsymbol{i}\boldsymbol{j}} \left( \mathbf{3} \, \text{Re}\,\boldsymbol{\Delta}\_{\boldsymbol{s}\mathbf{i}} \, \text{Re}\,\boldsymbol{\Delta}\_{\boldsymbol{s}\boldsymbol{i}} + \text{Im}\,\boldsymbol{\Delta}\_{\boldsymbol{s}\mathbf{i}} \, \text{Im}\,\boldsymbol{\Delta}\_{\boldsymbol{s}\boldsymbol{i}} \right) \end{split} \tag{35}$$

$$\begin{split} \left(P\_{\vec{y}}^{n21}\left(\{\Delta\}\right)\right) & \equiv \left(\frac{\mathcal{\lambda}\_{\mathsf{s}}}{\mathcal{\lambda}\_{\mathsf{d}}}\right)^{2} \sum\_{\boldsymbol{\upphi}\_{\mathsf{d}}} I\_{\boldsymbol{\upphi}\_{\mathsf{d}}\boldsymbol{\upphi}}\left(\operatorname{Re}\Delta\_{\mathsf{d}\_{\mathsf{l}}}\operatorname{Re}\Delta\_{\mathsf{d}\_{\mathsf{l}}} + 3\operatorname{Im}\Delta\_{\mathsf{d}\_{\mathsf{l}}}\operatorname{Im}\Delta\_{\mathsf{d}\_{\mathsf{l}}}\right) \\ & + 2\left(\frac{\mathcal{\lambda}\_{\mathsf{s}}}{\mathcal{\lambda}\_{\mathsf{d}}}\right)^{4} \sum\_{\boldsymbol{\upphi}\_{\mathsf{d}\_{\mathsf{l}}}} I\_{\boldsymbol{\upphi}\_{\mathsf{d}\_{\mathsf{l}}}}\left(\operatorname{Re}\Delta\_{\mathsf{s}\_{\mathsf{l}}}\operatorname{Re}\Delta\_{\mathsf{s}\_{\mathsf{l}}} + 3\operatorname{Im}\Delta\_{\mathsf{s}\_{\mathsf{l}}}\operatorname{Im}\Delta\_{\mathsf{s}\_{\mathsf{l}}}\right) \end{split} \tag{36}$$

$$\underline{Q}\_{\boldsymbol{\upbeta}}^{w}\left(\{\mathbf{A}\}\right) \equiv -\frac{3}{2} \underline{\xi}\_{d}^{2} \left(\Delta\_{d}^{0}\right)^{2} \left(\frac{\mathcal{A}\_{s}}{\mathcal{A}\_{\boldsymbol{\upbeta}}}\right)^{2} \sum\_{\stackrel{\scriptstyle \mathcal{A}\_{i}}{\longleftrightarrow}} \left(J\_{\stackrel{\scriptstyle \mathcal{A}\_{i}}{\longleftrightarrow}}^{\boldsymbol{\upalpha}} - J\_{\stackrel{\scriptstyle \mathcal{A}\_{i}}{\longleftrightarrow}}^{\boldsymbol{\upalpha}}\right) A\_{\boldsymbol{\upbeta}}^{\boldsymbol{\upalpha}}\tag{37}$$

$$\mathcal{Q}\_{\boldsymbol{\vartheta}}^{\boldsymbol{u},2}\left(\{\boldsymbol{\Delta}\}\right) \equiv \left(\frac{\mathcal{\mathcal{A}}\_{\boldsymbol{s}}}{\mathcal{\mathcal{A}}\_{\boldsymbol{d}}}\right)^{2} \sum\_{\boldsymbol{i}\_{\boldsymbol{i}\_{1}} \neq \boldsymbol{\mathcal{Y}}} I\_{\boldsymbol{i}\_{1} \boldsymbol{i}\_{2} \boldsymbol{\mathcal{Y}}} \, \operatorname{Im} \Delta\_{\operatorname{d}\_{\boldsymbol{\vartheta}}} \operatorname{Re} \Delta\_{\operatorname{d}\_{\boldsymbol{\vartheta}}} + 2 \left(\frac{\mathcal{\mathcal{A}}\_{\boldsymbol{s}}}{\mathcal{\mathcal{A}}\_{\boldsymbol{d}}}\right)^{4} \sum\_{\boldsymbol{i}\_{\boldsymbol{\vartheta}} \neq \boldsymbol{\mathcal{Y}}} I\_{\boldsymbol{i}\_{\boldsymbol{\vartheta}} \boldsymbol{i}\_{\boldsymbol{\vartheta}}} \, \operatorname{Im} \Delta\_{\operatorname{s}\_{\boldsymbol{\vartheta}}} \operatorname{Re} \Delta\_{\operatorname{s}\_{\boldsymbol{\vartheta}}} \tag{38}$$

$$V\_i^{sR} \left( \{ \Delta \} \right) \equiv 4 \left( \frac{\mathcal{A}\_s}{\mathcal{A}\_d} \right)^4 \sum\_{i\_1 i\_2 i\_3} I\_{i\_1 i\_2 i\_3} \Delta\_{s i\_1}^\* \operatorname{Re} \Delta\_{s i\_3} \tag{39}$$

$$V\_i^{sl} \left( \{ \Delta \} \right) \equiv 4 \left( \frac{\mathcal{\lambda}\_s}{\mathcal{\lambda}\_d} \right)^4 \sum\_{i\_1 i\_2 i\_3} I\_{i\_1 i\_2 i\_3} \Delta\_{s i\_1}^\* \operatorname{Im} \Delta\_{s i\_1} \tag{40}$$

For s-wave superconducting regions, they are defined similarly.

The coefficients in Eqs. 17 and 18 in the d-wave superconducting region are given as,

$$R\_y\left(\left\{\mathbf{A}\right\}\right) \equiv \frac{3}{2} \kappa\_d^2 \xi\_d^2 \left(\Delta\_d^0\right)^2 \left(\sum\_{a=x,y} K\_{y\_1}^{\text{car}}\right) + \frac{3}{4} \sum\_{i \neq j\_2} I\_{i\_l j\_2} \left[2\operatorname{Re}\left(\Delta\_{d\_l} \Delta\_{d\_2}^\*\right) + 4\left(\frac{\mathcal{A}\_s}{\mathcal{A}\_d}\right)^2 \operatorname{Re}\left(\Delta\_{s i\_l} \Delta\_{s i\_2}^\*\right)\right],\tag{41}$$

$$R\_{\boldsymbol{\vartheta}}^{2}\left(\left\{\mathbf{A}\right\}\right) \equiv \mathfrak{Z} \sum\_{i\_{\boldsymbol{\vartheta}\_{1}}} I\_{i\_{\boldsymbol{\vartheta}}, \boldsymbol{\vartheta}} \left[ \left(\frac{\mathcal{A}\_{s}}{\mathcal{A}\_{d}}\right) \mathrm{Re}\left(\boldsymbol{\Delta}\_{d\boldsymbol{\vartheta}}^{\*} \boldsymbol{\Delta}\_{s\boldsymbol{\varrho}\_{1}}\right) \right],\tag{42}$$

Composite Structures of d-Wave and s-Wave Superconductors (d-Dot):

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Analysis Using Two-Component Ginzburg-Landau Equations 341

$$\Delta S\_{ij} \equiv \frac{3}{2} \kappa\_d^2 \xi^2 \left(\Delta\_d^0\right)^2 \left(K\_{\bar{y}}^{xy} - K\_{\bar{y}}^{yx}\right) \tag{43}$$

$$T\_i^{\alpha} \equiv \frac{3}{2} \sum\_{i\_l i\_l} J\_{i\_l i\_l}^{\alpha} \left[ \text{Im} \left( \Delta\_{d\_l}^\* \Delta\_{d\_l} \right) + 2 \left( \frac{\mathcal{A}\_s}{\mathcal{A}\_d} \right)^2 \text{Re} \left( \Delta\_{d\_l}^\* \Delta\_{s i\_l} \right) \right] \left( \alpha = x, y \right), \tag{44}$$

$$T\_i^{2\alpha} \equiv \frac{3}{2} \sum\_{i\_l i\_2} J\_{i\_l i\_l}^{\alpha} \left( \frac{\mathcal{\lambda}\_i}{\mathcal{\lambda}\_d} \right) \left[ \text{Im} \left( \Delta\_{d\_l}^\* \Delta\_{s i\_2} \right) + \text{Im} \left( \Delta\_{s i\_l}^\* \Delta\_{d i\_2} \right) \right] \left( \alpha = \text{x}, \text{y} \right), \tag{45}$$

$$U\_i^a \equiv \frac{3}{4} \kappa\_d^2 \xi\_d^2 \left(\Delta\_d^0\right)^2 \frac{2\pi}{\Phi\_0} H J\_i^a \left(\alpha = \mathbf{x}, \mathbf{y}\right). \tag{46}$$
