**2. Model: two components Ginzburg-Landau free energy**

In order to discuss the basic properties of the d-dot, especially to describe the appearance of spontaneous magnetic fluxes, we use the phenomenological Ginzburg-Landau (GL) theory. However, for anisotropic superconductors, such as the 2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* -wave high-Tc cuprate superconductors, their anisotropy cannot be treated by the simple GL theory. This is because using only up to quartic term and quadratic term of gradients of the single order parameters in free energy, there are no anisotropic terms. Anisotropy of the vortices in dwave superconductors within the phenomenological theory was treated by Ren et al., who used a two-component GL theory [12-15]. Here, two components mean the two components of the order parameter of with s and d symmetries. They derived the two-component GL free energy from the Gor'kov equations.

322 Superconductors – Materials, Properties and Applications

**Figure 3.** Schematic diagram of a d-dot.

following sections.

flux system and our d-dot system.

electromagnetic force.

as an Ising spin system, Kirtley et al. made a frustrated triangular lattice of π-rings [7]. In their systems, the π-rings were isolated and interacted with each other purely by the

In contrast to the previous approach for using the spontaneous half-flux quantum, we considered nano-sized d-wave superconductor embedded in an s-wave matrix, as shown in Fig. 3 [9-11]. We want to consider the whole d-dot system as a single element, not the individual half-quantum fluxes. As in the single half-quantum system, our d-dot has two degenerate states if the spontaneous magnetic fluxes appear, because the state with spontaneous magnetic fluxes under zero external field also breaks time-reversal symmetry. This property is independent from the shape, and the d-dot in any shape always has two degenerate stable states. Therefore, the d-dot as a whole can be considered as a single element with two level states and it might be used as a spin or a bit also as a qubit. It has better properties than those of single flux quantum element, which will be shown in

In the following, we first show a phenomenological superconducting theory, which describes the spontaneous magnetic fluxes in these composite structures, especially in d-dot systems and then we discuss the basic properties of this d-dot, based on this phenomenological theory. Also, we discuss the difference between a single half-quantum

In order to discuss the basic properties of the d-dot, especially to describe the appearance of spontaneous magnetic fluxes, we use the phenomenological Ginzburg-Landau (GL) theory. However, for anisotropic superconductors, such as the 2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* -wave high-Tc cuprate

**2. Model: two components Ginzburg-Landau free energy** 

Multi-components GL equations were used for exotic superconductors, e.g. heavy fermion superconductors[16-18] and Sr2RuO4[19,20]. Also, recently, two components GL equations are studied for two-band or two-gap superconductors, such as MgB2 [21,22].

The model by Ren.et al. especially emphasize the anisotropy of d-wave superconductivity. Therefore, we use the following two-component Ginzburg-Landau (GL) free energy for dwave superconductors;

$$\begin{split} \mathsf{F}\_{d}\left(\Delta\_{s},\Delta\_{d},\mathbf{A}\right) &= \int\_{\Omega} \left[\frac{3\alpha}{8}\mathscr{A}\_{d}\left[\left|\Delta\_{d}\right|^{2} - \frac{4\ln\left(T\_{cd}/T\right)}{3\alpha}\right]^{2} + \alpha\mathscr{A}\_{d}\left[\left|\Delta\_{s}\right|^{4} + 2\frac{\alpha\_{s}}{\alpha}\left|\Delta\_{s}\right|^{2}\right] \right. \\ &+ \frac{1}{4}\alpha\mathscr{A}\_{d}\mathscr{v}\_{F}^{2}\left[\left|\mathbf{III}\Delta\_{s}\right|^{2} + 2\left|\mathbf{III}\Delta\_{s}\right|^{2} + \left(\Pi\_{s}^{\*}\Delta\_{s}\Pi\_{s}\Delta\_{d}^{\*} - \Pi\_{s}^{\*}\Delta\_{s}\Pi\_{j}\Delta\_{d}^{\*} + \text{H.c.}\right)\right] \\ &+ \alpha\mathscr{A}\_{d}\left[2\left|\Delta\_{s}\right|^{2}\left|\Delta\_{d}\right|^{2} + \frac{1}{2}\left(\Delta\_{s}^{\*2}\Delta\_{d}^{2} + \Delta\_{s}^{\*2}\Delta\_{d}^{\*2}\right)\right] + \frac{1}{8\pi}\left[\left|\mathbf{h} - \mathbf{H}\right|^{2} + \left(\operatorname{div}\mathbf{A}\right)^{2}\right]\right] d\Omega \end{split} \tag{2}$$

where <sup>2</sup> Π= ∇− *<sup>e</sup> i c* **A** is a generalized momentum operator that is gauge invariant and ( ) ( )<sup>2</sup> 7 3 8 ζ α π<sup>=</sup> *<sup>T</sup>* . Δ*<sup>d</sup>* and <sup>Δ</sup>*<sup>s</sup>* are the d-wave and the s-wave components of the order parameter,

respectively. λ= ( ) 0 *d d V N* is the strengths of the coupling constants for the d-wave

interaction channel and 1 α λ + = *s d s d V <sup>V</sup>* . Here, *<sup>N</sup>* ( ) <sup>0</sup> is the density of states of electrons at the Fermi energy and *Vd* and *Vs* are interaction constants between electrons for d- and s-wave channels, respectively. We assume attractive and repulsive interactions for the d- and the swave channels, respectively. *Tcd* is the transition temperature of the d-wave superconductivity under zero-external field. **H** is an external field and **h A** = curl . We take the London gauge i.e. ∇⋅ = **A** 0 and **A n**⋅ = 0 at the surface of superconductor. The term ( )<sup>2</sup> ∇ ⋅ **A** in the integrand of Eq. 2 is added for fixing the gauge.

Also, for s-wave superconductors, we use the following two-component GL equation with attractive and repulsive interactions for the s-wave and the d-wave channels, respectively:

$$\begin{split} \mathsf{F}\_{s}\left(\Delta\_{s},\Delta\_{d},\mathbf{A}\right) &= \int\_{\Omega} \left(\frac{\alpha}{2}\mathcal{A}\_{s}\left[\left|\Delta\_{s}\right|^{2} - \frac{\ln\left(T\_{\alpha}/T\right)}{\alpha}\right]^{2} + \frac{\alpha\mathcal{A}\_{s}}{2}\left[\frac{3}{8}\left|\Delta\_{d}\right|^{4} + \frac{\alpha\_{d}}{\alpha}\left|\Delta\_{d}\right|^{2}\right] \right. \\ &+ \frac{1}{8}\alpha\mathcal{A}\_{s}\upsilon\_{F}^{2}\left[\left|\mathbf{H}\Delta\_{d}\right|^{2} + 2\left|\mathbf{H}\Delta\_{s}\right|^{2} + \left(\Pi\_{s}^{\*}\Delta\_{s}\Pi\_{s}\Delta\_{d}^{\*} - \Pi\_{p}^{\*}\Delta\_{s}\Pi\_{p}\Delta\_{d}^{\*} + \text{H.c.}\right)\right] \\ &+ \frac{\alpha\mathcal{A}\_{s}}{2}\left[2\left|\Delta\_{s}\right|^{2}\left|\Delta\_{d}\right|^{2} + \frac{1}{2}\left(\Delta\_{s}^{\*2}\Delta\_{d}^{2} + \Delta\_{s}^{2}\Delta\_{d}^{\*2}\right)\right] + \frac{1}{8\varpi}\left[\left|\mathbf{h} - \mathbf{H}\right|^{2} + \left(\nabla\cdot\mathbf{A}\right)^{2}\right]\right]d\Omega \end{split} \tag{3}$$

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

*i jk k j a xy xy* (5)

*i jk by y* (6)

*ikj cxx* (7)

*N xy* (8)

*e ee N xy i j j ij* (9)

*x y N xy* (10)

*i i x y* is the coordinate of the i-th

*e e*

*<sup>e</sup> S* is an area of e-th element and the coefficients are defined as,

where ( ) *i jk* , , is a cyclic permutation of ( ) 1,2,3 and ( ) ,

( ) 1,2,3 . And then they have following properties;

normalized area of triangular elements.

= − *ee ee*

= −*e e*

= −*e e*

nodes of the e-th element (see Fig. 4(b)).These area coordinates are localized functions as shown in Fig. 5.Also the area coordinate ( ) , *<sup>e</sup> Ni x y* represents normalized area of a triangular of which nodes are P, Pj and Pk in Fig. 5 (b), where ( ) *i jk* , , is a cyclic permutation of

> ( ) <sup>3</sup> 1

 = *<sup>e</sup> <sup>i</sup> <sup>i</sup>*

( ) , =

=

**Figure 5.** (a) A bird's-eye view of an area coordinate ( ) , *<sup>e</sup> Ni x y* . (b) Area coordinates represent

() () <sup>3</sup> 1 , , = Δ =Δ *e e d di i e i*

Using these properties, the order parameters are expanded as

, 1

δ

Analysis Using Two-Component Ginzburg-Landau Equations 325

Here λ= ( ) 0 *s s V N* is the strengths of the coupling constants for the s-wave interaction

channeland 2 1 α λ − = *s d d s V <sup>V</sup>* .

In these free energies, the anisotropy of the d-wave superconductivity appears in the coupling terms of the gradient of both components of order parameters, ( ) H.c. ∗ ∗∗ ∗ Π Δ Π Δ −Π Δ Π Δ + *xs xd ys yd* , where the two terms that contain the gradients along the *x* and the *y* directions have different signs.

**Figure 4.** (a) Triangle elements of superconductors. (b) Nodes of a triangle element and the value of physical quantities at the nodes.

By minimizingthe sum of these free energies, we obtain the Ginzburg-Landau (GL) equations. For this purpose, we use the finite-element method [23-25], because we want to investigate variously shaped d-dots. Hereafter we consider two-dimensional system and ignore the variation of physical quantities along the direction perpendicular to the two dimensional system. In the finite-element method for two-dimensional system, we divide the superconductor region into small triangular elements (see Fig. 4 (a)) and then expand the order parameters and the vector potential in each element by using the area coordinate, which is defined as,

$$N\_{\iota}^{\circ}\left(\mathbf{x},\mathbf{y}\right) = \begin{cases} \frac{1}{2S\_{\iota}}\left(a\_{\iota} + b\_{\iota}\mathbf{x} + c\_{\iota}\mathbf{y}\right) & \text{inside of element} \\ 0 & \text{outside of element} \end{cases} \tag{4}$$

*<sup>e</sup> S* is an area of e-th element and the coefficients are defined as,

324 Superconductors – Materials, Properties and Applications

**A**

8

α

*d*

physical quantities at the nodes.

which is defined as,

*e ii i*

 

*i e*

=

Here

λ

channeland

F

αλ

αλ

*v*

2 1

*s d*

*V*

−

*<sup>V</sup>* .

λ

and the *y* directions have different signs.

*s*

*s*

( ) ( )

λ

α

ln <sup>3</sup> , , <sup>2</sup> 2 8

1 1 <sup>2</sup> 22 8

**Figure 4.** (a) Triangle elements of superconductors. (b) Nodes of a triangle element and the value of

By minimizingthe sum of these free energies, we obtain the Ginzburg-Landau (GL) equations. For this purpose, we use the finite-element method [23-25], because we want to investigate variously shaped d-dots. Hereafter we consider two-dimensional system and ignore the variation of physical quantities along the direction perpendicular to the two dimensional system. In the finite-element method for two-dimensional system, we divide the superconductor region into small triangular elements (see Fig. 4 (a)) and then expand the order parameters and the vector potential in each element by using the area coordinate,

> ( ) ( ) ( ) 1 inside of element , <sup>2</sup> i 1,2,3

*a bx cy N xy <sup>S</sup>* (4)

+ + = =

0 outside of element

*s d s d sd*

<sup>1</sup> <sup>2</sup> H.c.

+ Δ + Δ + Π Δ Π Δ −Π Δ Π Δ +

∗ ∗

*sF d s xs xd ys yd*

<sup>2</sup> <sup>2</sup> <sup>2</sup>

Ω

( )

Π Π (3)

2 4 2

 α

> α

**hH A**

*d*

αλ

( ) ( )

π

2 2 2 2 22 2 2

= ( ) 0 *s s V N* is the strengths of the coupling constants for the s-wave interaction

α

*T T*

*cs s d*

ΔΔ = Δ − + Δ + Δ

∗ ∗ ∗∗ ∗

*s sd s s d d*

+ Δ Δ + Δ Δ +Δ Δ + − + ∇⋅ Ω

In these free energies, the anisotropy of the d-wave superconductivity appears in the coupling terms of the gradient of both components of order parameters, ( ) H.c. ∗ ∗∗ ∗ Π Δ Π Δ −Π Δ Π Δ + *xs xd ys yd* , where the two terms that contain the gradients along the *x*

2

$$\mathbf{a}\_{i} = \mathbf{x}\_{j}^{e}\mathbf{y}\_{k}^{e} - \mathbf{x}\_{k}^{e}\mathbf{y}\_{j}^{e} \tag{5}$$

$$b\_i = \mathbf{y}\_j^e - \mathbf{y}\_k^e \tag{6}$$

$$\mathbf{x}\_{i} = \mathbf{x}\_{k}^{e} - \mathbf{x}\_{j}^{e} \tag{7}$$

where ( ) *i jk* , , is a cyclic permutation of ( ) 1,2,3 and ( ) , *e e i i x y* is the coordinate of the i-th nodes of the e-th element (see Fig. 4(b)).These area coordinates are localized functions as shown in Fig. 5.Also the area coordinate ( ) , *<sup>e</sup> Ni x y* represents normalized area of a triangular of which nodes are P, Pj and Pk in Fig. 5 (b), where ( ) *i jk* , , is a cyclic permutation of ( ) 1,2,3 . And then they have following properties;

$$\sum\_{l=1}^{3} N\_l^{\epsilon} \left( \mathbf{x}, \boldsymbol{\nu} \right) = \mathbf{l} \tag{8}$$

$$N\_i^\*\left(\mathbf{x}\_{\slash}^\*, \mathbf{y}\_{\not\equiv}^\*\right) = \delta\_{\not\equiv} \tag{9}$$

**Figure 5.** (a) A bird's-eye view of an area coordinate ( ) , *<sup>e</sup> Ni x y* . (b) Area coordinates represent normalized area of triangular elements.

Using these properties, the order parameters are expanded as

$$\Delta\_d\left(\mathbf{x}, \mathbf{y}\right) = \sum\_{\epsilon} \sum\_{i=1}^{3} \Delta\_{di}^{\epsilon} N\_i^{\epsilon}\left(\mathbf{x}, \mathbf{y}\right) \tag{10}$$

$$\Delta\_s\left(\mathbf{x}, \boldsymbol{\mathcal{y}}\right) = \sum\_{\boldsymbol{\epsilon}} \sum\_{i=1}^{\boldsymbol{\beta}} \Delta\_{si}^{\boldsymbol{\epsilon}} N\_i^{\boldsymbol{\epsilon}}\left(\mathbf{x}, \boldsymbol{\mathcal{y}}\right) \tag{11}$$

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

Here coefficients are also given in Appendix. At the boundary of d- and s-wave superconductors, because the wave function is continuous, following boundary conditions

> 1 2 1 2 Δ Δ <sup>=</sup>*<sup>s</sup> <sup>s</sup> V V <sup>s</sup> <sup>s</sup>*

1 2 1 2 Δ Δ <sup>=</sup> *d d V V d d*

In this model, d- and s-wave components of the order parameter interfere anisotropically with each other in the both of d-wave and s-wave superconductors. And the boundaries between both superconductors are assumed clean. But we cannot take into account the

**Figure 6.** Mesh partition used in the finite element method. Red, blue and grey regions are d-wave and

For treating roughness of the junctions, we consider second model. In contrast to the first model, in the second model, the s-wave superconductor is connected to the d-wave superconductor through a thin metal or an insulator layers. In the second model, we consider only the d-wave (s-wave) component of the order parameter in the d-wave (swave) superconducting region, but in the thin metal layer, we take both components. So,

( ) ( ) 2

( ) ( ) <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> ln 1 1 1 , div 2 4 88

αλ

Δ = Δ− + Δ + − + <sup>Ω</sup> *cs*

*T T* <sup>F</sup> **<sup>A</sup>** *v d* <sup>Π</sup> **hH A** (22)

α

α

*d d d d dF d*

*s s s s sF s*

<sup>2</sup> <sup>2</sup> 2 2 <sup>3</sup> 4ln 1 <sup>2</sup> 1 1 , div 8 34 8 8

αλ

*T T* <sup>F</sup> **<sup>A</sup>** *v d* <sup>Π</sup> **hH A** (21)

Δ = Δ− + Δ+ − + Ω *cd*

π

π

∗

∗

 π

 π

s-wave superconducting and junction regions, respectively.

are applied:

roughness of the boundary.

free energies are given as,

α λ

> α λ

Ω

Ω

Analysis Using Two-Component Ginzburg-Landau Equations 327

(19)

(20)

and also the magnetic vector potential is expanded as,

$$\mathbf{A}\left(\mathbf{x},\boldsymbol{\mathcal{y}}\right) = \sum\_{\boldsymbol{\epsilon}} \sum\_{i=1}^{3} \mathbf{A}\_{i}^{\epsilon} N\_{i}^{\epsilon}\left(\mathbf{x},\boldsymbol{\mathcal{y}}\right) \tag{12}$$

where Δ*<sup>e</sup> di* and Δ*<sup>e</sup> si* are the value of the d-wave and the s-wave order parameters at the i-th vertex of e-th element, respectively (see Fig. 4 (b)). Also, *<sup>e</sup>* **A***i* is the value of the vector potential at the i-th vertex of the e-th element (see Fig. 4 (b)). Inserting these expansions into the free energies Eqs. 2 and 3, the free energies are expressed by those values, Δ*<sup>e</sup> di* , Δ*<sup>e</sup> si* and *<sup>e</sup>* **A***<sup>i</sup>* . Then, minimizing the free energies, we get following GL equations. For the d-wave superconducting order parameter,

$$\begin{aligned} &\sum\_{j} \left[ P\_{y}^{dd} \left( \{A\} \right) + P\_{y}^{dt2R} \left( \{\Delta\} \right) \right] \text{Re}\,\Delta\_{\boldsymbol{\vartheta}} + \sum\_{j} \left[ Q\_{y}^{dd} \left( \{A\} \right) + Q\_{y}^{d2} \left( \{\Delta\} \right) \right] \text{Im}\,\Delta\_{\boldsymbol{\vartheta}} \\ &+ \sum\_{j} P\_{y}^{ds} \left( \{A\} \right) \text{Re}\,\Delta\_{\boldsymbol{\vartheta}} + \sum\_{j} Q\_{y}^{ds} \left( \{A\} \right) \text{Im}\,\Delta\_{\boldsymbol{\vartheta}} = V\_{i}^{d\mathcal{R}} \left( \{\Delta\} \right) \end{aligned} \tag{13}$$
 
$$\begin{aligned} &\sum\_{j} \left[ -Q\_{y}^{d\mathcal{I}} \left( \{A\} \right) + Q\_{y}^{d\mathcal{I}2} \left( \{\Delta\} \right) \right] \text{Re}\,\Delta\_{\boldsymbol{\vartheta}} + \sum\_{j} \left[ P\_{y}^{d\mathcal{I}} \left( \{A\} \right) + P\_{y}^{d\mathcal{I}2} \left( \{\Delta\} \right) \right] \text{Im}\,\Delta\_{\boldsymbol{\vartheta}} \\ &+ \sum\_{j} P\_{y}^{d\mathcal{I}} \left( \{A\} \right) \text{Im}\,\Delta\_{\boldsymbol{\vartheta}} - \sum\_{j} Q\_{y}^{d\mathcal{I}} \left( \{A\} \right) \text{Re}\,\Delta\_{\boldsymbol{\vartheta}} = V\_{i}^{dl} \end{aligned} \tag{14}$$

And for the s-wave superconducting order parameter,

$$\begin{aligned} \sum\_{j} \left[ P\_{\vec{y}}^{\text{as}} \left( \{A\} \right) + P\_{\vec{y}}^{\text{as}2R} \left( \{\Delta\} \right) \right] \text{Re}\,\Delta\_{\vec{y}} + \sum\_{j} \left[ \mathcal{Q}\_{\vec{y}}^{\text{as}} \left( \{A\} \right) + \mathcal{Q}\_{\vec{y}}^{\text{as}2} \left( \{\Delta\} \right) \right] \text{Im}\,\Delta\_{\vec{y}} \\ + \sum\_{j} P\_{\vec{y}}^{\text{as}} \left( \{A\} \right) \text{Re}\,\Delta\_{\vec{y}} + \sum\_{j} \mathcal{Q}\_{\vec{y}}^{\text{as}} \left( \{A\} \right) \text{Im}\,\Delta\_{\vec{y}} = V\_{\vec{\imath}}^{\text{as}} \left( \{\Delta\} \right) \end{aligned} \tag{15}$$

$$\begin{aligned} &\sum\_{j} \left[ -\mathcal{Q}\_{\vec{y}}^{w} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} + \mathcal{Q}\_{\vec{y}}^{w2} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} \right] \text{Re}\,\Delta\_{\vec{y}} + \sum\_{j} \left[ P\_{\vec{y}}^{w} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} + P\_{\vec{y}}^{w2I} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} \right] \text{Im}\,\Delta\_{\vec{y}} \\ &+ \sum\_{j} P\_{\vec{y}}^{w} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} \text{Im}\,\Delta\_{\vec{y}} - \sum\_{j} \mathcal{Q}\_{\vec{y}}^{w} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} \text{Re}\,\Delta\_{\vec{y}} = V\_{\wedge}^{w} \left\{ \begin{bmatrix} A \end{bmatrix} \right\} \end{aligned} \tag{16}$$

Coefficients for d-wave region and s-wave region are different and are given in Appendix. Also for the vector potential following equations are obtained as follows,

$$\sum\_{j} \left[ R\_{\circ}^{e} \left( \left\{ \Delta\_{d}, \Delta\_{s} \right\} \right) + R\_{\circ}^{2e} \left( \left\{ \Delta\_{d}, \Delta\_{s} \right\} \right) \right] A\_{j\times} + \sum\_{j} S\_{\circ}^{e} A\_{j\circ} = T\_{i}^{e\alpha} - T\_{i}^{2\alpha} - U\_{i}^{s\gamma} \tag{17}$$

$$\sum\_{j} \left[ R^{e}\_{\bar{y}} \left( \{ \Delta\_{d}, \Delta\_{s} \} \right) - R^{2e}\_{\bar{y}} \left( \{ \Delta\_{d}, \Delta\_{s} \} \right) \right] A\_{\bar{y}} - \sum\_{j} S^{e}\_{\bar{y}} A\_{\bar{y}} = T^{e\gamma}\_{i} + T^{2\alpha}\_{i} - U^{\alpha}\_{i} \tag{18}$$

Here coefficients are also given in Appendix. At the boundary of d- and s-wave superconductors, because the wave function is continuous, following boundary conditions are applied:

326 Superconductors – Materials, Properties and Applications

where Δ*<sup>e</sup>*

*di* and Δ*<sup>e</sup>*

superconducting order parameter,

and also the magnetic vector potential is expanded as,

() () <sup>3</sup> 1 , , = Δ =Δ *e e s si i e i*

() () <sup>3</sup> 1 , , = <sup>=</sup> *e e i i*

vertex of e-th element, respectively (see Fig. 4 (b)). Also, *<sup>e</sup>* **A***i* is the value of the vector potential at the i-th vertex of the e-th element (see Fig. 4 (b)). Inserting these expansions into

Then, minimizing the free energies, we get following GL equations. For the d-wave

{ } ( ) { } ( ) { } ( ) { } ( )

+ Δ Δ+ + ΔΔ

( ) { } ( ) { } ( ) { } ( ) { }

− + Δ Δ+ + Δ Δ

( ) { } ( ) { } ( ) { } ( ) { }

*ij ij sj ij ij sj j j*

( ) { } ( ) { } ( ) { } ( ) { }

− + Δ Δ+ + Δ Δ

Coefficients for d-wave region and s-wave region are different and are given in Appendix.

( ) { } ( ) { } 2 2 ΔΔ + ΔΔ + = − − , , *<sup>e</sup> <sup>e</sup> <sup>e</sup> ex ex ey ij d s ij d s jx ij jy i i i*

( ) { } ( ) { } 2 2 ΔΔ − ΔΔ − = + − , , *<sup>e</sup> <sup>e</sup> <sup>e</sup> ey ey ex ij d s ij d s jy ij jx i i i*

+ Δ Δ+ + Δ Δ

( ) { } ( ) { } ( ) { }

*ss ss ss ss I ij ij sj ij ij sj j j*

( ) { } ( ) { } ( ) { }

*QAQ PA P*

*ss ss R ss ss*

*PA P QAQ*

Re Im

Im Re

*ds ds sI ij dj ij dj i j j*

+ Δ − Δ= Δ

Also for the vector potential following equations are obtained as follows,

*j j*

*j j*

+ Δ + Δ= Δ

*ds ds sR ij dj ij dj i j j*

2 2 Re Im

*dd dd dd dd I ij ij dj ij ij dj j j*

*Q AQ PAP*

*ij ij dj ij ij dj*

the free energies Eqs. 2 and 3, the free energies are expressed by those values, Δ*<sup>e</sup>*

{ } ( ) { } ( ) { } ( )

*dd dd R dd dd*

*PAP Q AQ*

Re Im

*ds ds dR ij sj ij sj i*

+ Δ + Δ= Δ

Im Re

*ds ds dI ij sj ij sj i j j*

*j j*

( ) { } ( ) { }

And for the s-wave superconducting order parameter,

+ Δ − Δ =

*j j*

*si* are the value of the d-wave and the s-wave order parameters at the i-th

2 2 Re Im

*PA QA V* (13)

*PA QA V* (14)

2 2 Re Im

2 2 Re Im

*PA QA V* (15)

*PA QA V* (16)

*R R A SA T T U* (17)

*R R A SA T T U* (18)

*e i*

*x y N xy* (11)

**A A** *x y N xy* (12)

*di* , Δ*<sup>e</sup>*

*si* and *<sup>e</sup>* **A***<sup>i</sup>* .

$$\frac{\Delta\_{1s}}{V\_{1s}} = \frac{\Delta\_{2s}}{V\_{2s}}\tag{19}$$

$$\frac{\Delta\_{1d}}{V\_{1d}} = \frac{\Delta\_{2d}}{V\_{2d}}\tag{20}$$

In this model, d- and s-wave components of the order parameter interfere anisotropically with each other in the both of d-wave and s-wave superconductors. And the boundaries between both superconductors are assumed clean. But we cannot take into account the roughness of the boundary.

**Figure 6.** Mesh partition used in the finite element method. Red, blue and grey regions are d-wave and s-wave superconducting and junction regions, respectively.

For treating roughness of the junctions, we consider second model. In contrast to the first model, in the second model, the s-wave superconductor is connected to the d-wave superconductor through a thin metal or an insulator layers. In the second model, we consider only the d-wave (s-wave) component of the order parameter in the d-wave (swave) superconducting region, but in the thin metal layer, we take both components. So, free energies are given as,

$$\mathbb{E}\_d\left(\Delta\_d, \mathbf{A}\right) = \int\_{\Omega} \left(\frac{3\alpha}{8}\mathcal{A}\_d \left[\left|\Delta\_d\right|^2 - \frac{4\ln T\_{cd}/T}{3\alpha}\right]^2 + \frac{1}{4}\alpha\mathcal{Q}\_d\mathbf{v}\_F^2 \left|\mathbf{H}\Delta\_d^\*\right|^2 + \frac{1}{8\pi}\left|\mathbf{h} - \mathbf{H}\right|^2 + \frac{1}{8\pi}\left(\text{div}\,\mathbf{A}\right)^2\right) d\Omega \tag{21}$$

$$\mathbb{E}\_s\left(\Delta\_s, \mathbf{A}\right) = \int\_{\Omega} \left(\frac{\alpha}{2}\lambda\_s \left[\left|\Delta\_s\right| - \frac{\ln T\_{cs}/T}{\alpha}\right]^2 + \frac{1}{4}\alpha\lambda\_s\mathbf{v}\_F^2 \left|\mathbf{H}\Delta\_s^\*\right|^2 + \frac{1}{8\pi}\left|\mathbf{h} - \mathbf{H}\right|^2 + \frac{1}{8\pi}\left(\text{div}\,\mathbf{A}\right)^2\right)d\Omega \tag{22}$$

$$\begin{split} \mathbb{F}\_{\mathcal{M}}\left(\Delta\_{s},\Delta\_{s'},\mathbf{A}\right) &= \int\_{\Omega} \left| \mathcal{J}\_{s} \right| \frac{3b}{2} \left[ \left| \Delta\_{s} \right|^{2} + \frac{a\_{s}}{3b} \right]^{2} + b \left[ \left| \Delta\_{s} \right|^{2} + \frac{a\_{s}}{2b} \right]^{2} \left| \mathcal{J}\_{s} \right| + b \left| \mathcal{J}\_{s} \right| \left[ 2 \left| \Delta\_{s} \right|^{2} \left| \Delta\_{s} \right|^{2} + \frac{1}{2} \left( \Delta\_{s}^{\*2} \Delta\_{s}^{2} + \Delta\_{s}^{\*2} \Delta\_{s}^{\*2} \right) \right. \\ &\left. + \frac{1}{4} \nu\_{\rho}^{2} \left[ \left| \mathbf{H} \Delta\_{s} \right|^{2} + 2 \left| \mathbf{H} \Delta\_{s} \right|^{2} + \left( \Pi\_{s}^{\*} \Delta\_{s} \Pi\_{s} \Delta\_{s}^{\*} - \Pi\_{\rho}^{\*} \Delta\_{s} \Pi\_{\rho} \Delta\_{s}^{\*} + \text{H.c.} \right) \right] + \frac{1}{8\pi} \left| \mathbf{h} - \mathbf{H} \right|^{2} + \frac{1}{8\pi} \left( \text{div} \, \mathbf{A} \right)^{2} \right] d\Omega \end{split} \tag{23}$$

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

ξ

In this calculation, the physical parameters, such as the GL parameters and the transition temperatures, for Nb and YBCO in the s- and the d-wave superconducting regions are used,

ξ

superconductors at zero temperature. In Fig.7 (c), right-lower part is the d-wave superconductor and other part is s-wave superconductor. Both components of the order parameter exist in both of s- and d-wave superconductors. Especially, s-wave component penetrates into the d-wave superconducting region and interferes with d-wave component. Such interference causes the spontaneous magnetic field, which is perpendicular to the plane. In Fig. 7 (c), blue color means magnetic field *Hz* is positive. Total magnetic flux around the corner is approximately <sup>0</sup> Φ 2 . So a half-quantum magnetic flux appears spontaneously. In this figure, the size of d-wave superconductor is 0 0 10 10

the size of d-wave region is 40 nm, which is rather small compare to the experiments by Hilgencamp et al. [6]. But for such nano-sized d-wave superconductors, half-quantum magnetic fluxes appear, as this result shows. For larger d-wave region, the spontaneous

**Figure 8.** Superconducting order parameter and magnetic field structures for a square d-wave superconductor in an s-wave superconductor. (a) Amplitude of d-wave component of the order parameter. (b) Amplitude of s-wave component of the order parameter. (c) Magnetic field distribution.

ξ*L <sup>d</sup>* . Here, <sup>0</sup>

coherence length of high-Tc superconductors is rather small ( <sup>0</sup>

and system size is set as <sup>0</sup> = 25

magnetic flux is expected to appear easily.

Blue means *H z* < 0.

Analysis Using Two-Component Ginzburg-Landau Equations 329

*<sup>d</sup>* is the coherence length of the d-wave

ξ×

= 2~4 *<sup>d</sup>* nm for YBCO) and

ξ*d d* . The

Here ( ) Δ , F*d d* **A** , ( ) Δ , F*s s* **A** and ( ) Δ Δ, , F*M sd* **A** are the free energies for d-wave, s-wave superconducting and junction regions respectively. A typical division of superconducting region to d- and s-wave superconducting and junction regions is given in Fig. 6. In the junction region, order parameters and the vector potential vary rapidly and therefore smaller mesh sizes are taken. At the boundary, we assume that each component of the order parameter is continuous.

## **3. Appearance of the half-quantum magnetic flux in d-dot**

Using the two models in Section 2, the spontaneous magnetic flux can be described. In Fig.7, the s-wave and d-wave order parameters and magnetic field distribution are shown for a corner junction between s-wave and d-wave superconductors under zero external magnetic field.

**Figure 7.** Appearance of spontaneous half-quantum magnetic flux at a corner junction. (a) Amplitude of s-wave component of order parameter. (b) Amplitude of s-wave component of order parameter. (c) Magnetic field distribution at zero external field.

In this calculation, the physical parameters, such as the GL parameters and the transition temperatures, for Nb and YBCO in the s- and the d-wave superconducting regions are used, and system size is set as <sup>0</sup> = 25ξ *L <sup>d</sup>* . Here, <sup>0</sup> ξ *<sup>d</sup>* is the coherence length of the d-wave superconductors at zero temperature. In Fig.7 (c), right-lower part is the d-wave superconductor and other part is s-wave superconductor. Both components of the order parameter exist in both of s- and d-wave superconductors. Especially, s-wave component penetrates into the d-wave superconducting region and interferes with d-wave component. Such interference causes the spontaneous magnetic field, which is perpendicular to the plane. In Fig. 7 (c), blue color means magnetic field *Hz* is positive. Total magnetic flux around the corner is approximately <sup>0</sup> Φ 2 . So a half-quantum magnetic flux appears spontaneously. In this figure, the size of d-wave superconductor is 0 0 10 10 ξ × ξ *d d* . The coherence length of high-Tc superconductors is rather small ( <sup>0</sup> ξ = 2~4 *<sup>d</sup>* nm for YBCO) and the size of d-wave region is 40 nm, which is rather small compare to the experiments by Hilgencamp et al. [6]. But for such nano-sized d-wave superconductors, half-quantum magnetic fluxes appear, as this result shows. For larger d-wave region, the spontaneous magnetic flux is expected to appear easily.

328 Superconductors – Materials, Properties and Applications

Ω

**A**

Π Π

F

parameter is continuous.

λ

∗ ∗ ∗ ∗∗ ∗

*F d s xs xd ys yd*

*d s*

**3. Appearance of the half-quantum magnetic flux in d-dot** 

( ) ( )

Δ Δ = Δ + + Δ + + Δ Δ + Δ Δ +Δ Δ

Here ( ) Δ , F*d d* **A** , ( ) Δ , F*s s* **A** and ( ) Δ Δ, , F*M sd* **A** are the free energies for d-wave, s-wave superconducting and junction regions respectively. A typical division of superconducting region to d- and s-wave superconducting and junction regions is given in Fig. 6. In the junction region, order parameters and the vector potential vary rapidly and therefore smaller mesh sizes are taken. At the boundary, we assume that each component of the order

Using the two models in Section 2, the spontaneous magnetic flux can be described. In Fig.7, the s-wave and d-wave order parameters and magnetic field distribution are shown for a corner junction between s-wave and d-wave superconductors under zero external magnetic field.

**Figure 7.** Appearance of spontaneous half-quantum magnetic flux at a corner junction. (a) Amplitude of s-wave component of order parameter. (b) Amplitude of s-wave component of order parameter. (c)

Magnetic field distribution at zero external field.

*M sd d d s d d s d s d sd*

23 2 2

λ

2 2 2 2 22 2 2

π

 π

**hH A**

∗ ∗

(23)

 λ

{ } ( ) ( )

*v d*

+ Δ + Δ + Π Δ Π Δ −Π Δ Π Δ + + − + <sup>Ω</sup>

2 2 2 2 <sup>2</sup>

2 2

3 1 , , <sup>2</sup>

*ba a b b b b*

<sup>1</sup> 1 1 <sup>2</sup> H.c. div 4 8 8

 

**Figure 8.** Superconducting order parameter and magnetic field structures for a square d-wave superconductor in an s-wave superconductor. (a) Amplitude of d-wave component of the order parameter. (b) Amplitude of s-wave component of the order parameter. (c) Magnetic field distribution. Blue means *H z* < 0.

How such spontaneous magnetic flux appears, when a square d-wave superconductor is embedded in an s-wave superconductor? In Fig.8, the typical magnetic flux structure of such square d-dot in which edge of square is parallel to the x- or y-directions, under zero magnetic field is shown. In this figure, red color means *H z* < 0. Magnetic fluxes appear at four corners of the square d-dot, and they order antiferromagnetically. Such antiferromagnetic order is already obtained by the experiment for zigzag junctions by Hilgencamp et al. [6]. The reason of this antiferromagnetic order can be understood by considering the interaction between vortices. Usual theory of vortex interaction tells us that parallel vortices repel each other and antiparallel vortices attract each other because of the current around the vortices and Lorentz force by this current. For the d-dot in Fig.8, the spontaneous current flows form top-center of the d-wave superconductors, turns right and left and flows into left and right sides of d-wave superconductor. So this current naturally creates upward flux at the left-top corner and downward flux at the right-top corner. Fluxes at the lower two corners can be explained similarly.

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

The second model of d-dots (Eqs.21-23) also describes these spontaneous half-quantum magnetic fluxes. Using this model, we argue the size-dependence of these magnetic fluxes. Question is: The d-dot in Fig.8 have four half-quantum magnetic fluxes, but if the size of ddot becomes small, then what happens? In Fig.9, the typical size dependences of the order parameter structure and the magnetic fluxes are shown. In these figures, for smaller d-dots the amplitudes of d-wave order parameter and magnetic field distributions are shown. When the size of the d-dot becomes small, then the amplitude of d-wave component of the order parameter becomes small, and the spontaneous magnetic field becomes small. This is because the spontaneous current, which flows around each corner, does not decrease much at the center of the edge of the square when the size of the d-dot is comparable to the coherence length. Then, the total magnetic flux becomes less than the half-flux quantum,

The temperature dependences of amplitude of order parameters and the spontaneous magnetic fluxes also show similar tendency, because in this case, the coherence length

**Figure 10.** Order parameter ((a) d-wave and (b) s-wave) and magnetic field (c) distribution for an s-dot.

although the fluxoid is still a half-flux quantum.

increases with increasing temperature.

Analysis Using Two-Component Ginzburg-Landau Equations 331

**Figure 9.** The d-wave superconducting order parameter amplitudes ( (a) and (c)) and magnetic field distribution ( (b) and (c) ) for smaller d-dot with d=1.7ξ. ( (a) and (b)) with d=1.3 ξ ((c) and (d)).

The second model of d-dots (Eqs.21-23) also describes these spontaneous half-quantum magnetic fluxes. Using this model, we argue the size-dependence of these magnetic fluxes. Question is: The d-dot in Fig.8 have four half-quantum magnetic fluxes, but if the size of ddot becomes small, then what happens? In Fig.9, the typical size dependences of the order parameter structure and the magnetic fluxes are shown. In these figures, for smaller d-dots the amplitudes of d-wave order parameter and magnetic field distributions are shown. When the size of the d-dot becomes small, then the amplitude of d-wave component of the order parameter becomes small, and the spontaneous magnetic field becomes small. This is because the spontaneous current, which flows around each corner, does not decrease much at the center of the edge of the square when the size of the d-dot is comparable to the coherence length. Then, the total magnetic flux becomes less than the half-flux quantum, although the fluxoid is still a half-flux quantum.

330 Superconductors – Materials, Properties and Applications

at the lower two corners can be explained similarly.

How such spontaneous magnetic flux appears, when a square d-wave superconductor is embedded in an s-wave superconductor? In Fig.8, the typical magnetic flux structure of such square d-dot in which edge of square is parallel to the x- or y-directions, under zero magnetic field is shown. In this figure, red color means *H z* < 0. Magnetic fluxes appear at four corners of the square d-dot, and they order antiferromagnetically. Such antiferromagnetic order is already obtained by the experiment for zigzag junctions by Hilgencamp et al. [6]. The reason of this antiferromagnetic order can be understood by considering the interaction between vortices. Usual theory of vortex interaction tells us that parallel vortices repel each other and antiparallel vortices attract each other because of the current around the vortices and Lorentz force by this current. For the d-dot in Fig.8, the spontaneous current flows form top-center of the d-wave superconductors, turns right and left and flows into left and right sides of d-wave superconductor. So this current naturally creates upward flux at the left-top corner and downward flux at the right-top corner. Fluxes

**Figure 9.** The d-wave superconducting order parameter amplitudes ( (a) and (c)) and magnetic field distribution ( (b) and (c) ) for smaller d-dot with d=1.7ξ. ( (a) and (b)) with d=1.3 ξ ((c) and (d)).

The temperature dependences of amplitude of order parameters and the spontaneous magnetic fluxes also show similar tendency, because in this case, the coherence length increases with increasing temperature.

**Figure 10.** Order parameter ((a) d-wave and (b) s-wave) and magnetic field (c) distribution for an s-dot.

These spontaneous magnetic fluxes do not appears for the geometry for a square d-wave superconductor embedded in an s-wave superconducting matrix, but they appear for a square s-wave superconductor embedded in d-wave superconducting matrix and various shaped d-wave superconductors embedded in the s-wave superconducting matrix. The swave superconducting dot case is called "s-dot". For an s-dot, order parameter structures and field distribution are shown in Fig. 10. There appear spontaneous magnetic fluxes around the corners antiferromagnetically. These magnetic fluxes also are explained similarly by the phase anisotropy of d-wave superconductivity. But the distribution of magnetic field is opposite to the d-dot case, where magnetic field appears mainly outside of inner superconducting region, but in this case, magnetic field appears mainly inside of inner superconducting region. The s-dot is also useful but in the following we focus on the d-dot case, which is more easily fabricated, we think.

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

**Figure 12.** Spatial distribution of d-wave component of the order parameter (a) and magnetic field for a

**Figure 13.** Spatial distribution of magnetic field for a square d-dot. (a) *H* = 0 , (b) <sup>2</sup>

These doubly degenerate states have good properties for applications. First they remain under weak external magnetic field. In Fig. 13, external field dependence of spontaneous magnetic field distributions for a square d-dot is shown. The spontaneous magnetic flux parallel (anti-parallel) to the external field becomes large (small), respectively. Although the degeneracy from broken time reversal symmetry is lifted under the external magnetic field,

, and (e) <sup>2</sup>

<sup>0</sup> *H L* = Φ 0.2 2

π.

π

rotated square d-dot.

(c) <sup>2</sup> <sup>0</sup> *H L* = Φ 0.05 2

π

 , (d) <sup>2</sup> <sup>0</sup> *H L* = Φ 0.1 2

Analysis Using Two-Component Ginzburg-Landau Equations 333

<sup>0</sup> *H L* = Φ 0.02 2

π,

Next we discuss the shape dependence of the d-dot. Even if the shape of the d-wave superconducting region is different from the square that is parallel to the crystal axis or xand y-axis, the spontaneous magnetic field is also expected. In Fig. 11, distributions of the order parameters and magnetic field for an equilateral triangle plate are shown. For this case the spontaneous magnetic fluxes appear along the upper edges connected to the top corner. The spontaneous current flows the top corner and return to the intermediate points of upper edges. Also spontaneous magnetic fluxes appear around the lower corners. These spontaneous magnetic field along the edge also appear for rotated or diagonal squares, as shown in Fig. 12. In these figures, the spontaneous currents across the junction mainly flow along x- or y- directions. And direction of junction between d- and s-wave superconductors is important for the appearance of magnetic flux.

**Figure 11.** Spatial distribution of d-wave component of the order parameter (a) and magnetic field for a triangular d-dot.

Also we note that not only square d-dots that is parallel to the x- and y-axis but also arbitrary shaped d-dots that show spontaneous magnetic field, such as in Figs. 11 and 12, have doubly degenerate stable states. And the shape of d-dot controls 15 the magnetic field distribution.

Composite Structures of d-Wave and s-Wave Superconductors (d-Dot): Analysis Using Two-Component Ginzburg-Landau Equations 333

332 Superconductors – Materials, Properties and Applications

case, which is more easily fabricated, we think.

is important for the appearance of magnetic flux.

a triangular d-dot.

distribution.

These spontaneous magnetic fluxes do not appears for the geometry for a square d-wave superconductor embedded in an s-wave superconducting matrix, but they appear for a square s-wave superconductor embedded in d-wave superconducting matrix and various shaped d-wave superconductors embedded in the s-wave superconducting matrix. The swave superconducting dot case is called "s-dot". For an s-dot, order parameter structures and field distribution are shown in Fig. 10. There appear spontaneous magnetic fluxes around the corners antiferromagnetically. These magnetic fluxes also are explained similarly by the phase anisotropy of d-wave superconductivity. But the distribution of magnetic field is opposite to the d-dot case, where magnetic field appears mainly outside of inner superconducting region, but in this case, magnetic field appears mainly inside of inner superconducting region. The s-dot is also useful but in the following we focus on the d-dot

Next we discuss the shape dependence of the d-dot. Even if the shape of the d-wave superconducting region is different from the square that is parallel to the crystal axis or xand y-axis, the spontaneous magnetic field is also expected. In Fig. 11, distributions of the order parameters and magnetic field for an equilateral triangle plate are shown. For this case the spontaneous magnetic fluxes appear along the upper edges connected to the top corner. The spontaneous current flows the top corner and return to the intermediate points of upper edges. Also spontaneous magnetic fluxes appear around the lower corners. These spontaneous magnetic field along the edge also appear for rotated or diagonal squares, as shown in Fig. 12. In these figures, the spontaneous currents across the junction mainly flow along x- or y- directions. And direction of junction between d- and s-wave superconductors

**Figure 11.** Spatial distribution of d-wave component of the order parameter (a) and magnetic field for

Also we note that not only square d-dots that is parallel to the x- and y-axis but also arbitrary shaped d-dots that show spontaneous magnetic field, such as in Figs. 11 and 12, have doubly degenerate stable states. And the shape of d-dot controls 15 the magnetic field

**Figure 12.** Spatial distribution of d-wave component of the order parameter (a) and magnetic field for a rotated square d-dot.

**Figure 13.** Spatial distribution of magnetic field for a square d-dot. (a) *H* = 0 , (b) <sup>2</sup> <sup>0</sup> *H L* = Φ 0.02 2π , (c) <sup>2</sup> <sup>0</sup> *H L* = Φ 0.05 2π , (d) <sup>2</sup> <sup>0</sup> *H L* = Φ 0.1 2π , and (e) <sup>2</sup> <sup>0</sup> *H L* = Φ 0.2 2π.

These doubly degenerate states have good properties for applications. First they remain under weak external magnetic field. In Fig. 13, external field dependence of spontaneous magnetic field distributions for a square d-dot is shown. The spontaneous magnetic flux parallel (anti-parallel) to the external field becomes large (small), respectively. Although the degeneracy from broken time reversal symmetry is lifted under the external magnetic field,

the state with π/2 rotated magnetic field distributions are equally stable. These doubly degenerate states come from the broken four-fold symmetry of the square shape. This property depends on the shape of d-dots. For asymmetric shaped d-dots, one of the doubly degenerate states becomes more stable than another state. This means that we can control these degenerate states using the magnetic field for asymmetric d-dots.

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

As shown in previous section, the d-dots have double degenerate stable states. So we can use them as bits or 1/2 spins. In order to use them as artificial spins, the d-dots will be placed periodically or randomly. Then the interaction between them is important for these spin systems. For using the d-dots as computational bits, they are also placed to transform the information. Therefore interaction between d-dots also important for these applications.

How the d-dots interact with each other? Interaction between d-dots basically comes from the interaction between spontaneous magnetic fluxes or vortices. If the spontaneous vortices are independent from each other, that is, if there is no current flow between the vortices, then via a purely electromagnetic interaction, they interact. This is the case of the π-ring system of Kirtley et al. [4]. If the vortices are interacting in the same superconductors, there is a supercurrent flow around the vortices and ordinary vortices interact with each other through this current. The current distribution around a singly quantized vortex is given by the first order Bessel function. And therefore the interaction force to vortex 1 from vortex 2

> ( )( ) 0 0 1 2 <sup>12</sup> <sup>12</sup> 2 3 1 12 <sup>ˆ</sup> <sup>8</sup>πλ

Φ

Φ

 <sup>=</sup> *<sup>r</sup>* **f r** *<sup>K</sup>*

Where <sup>12</sup> *r* and <sup>12</sup> **r**ˆ are distance between two vortices and a unit vector from the vortex 2 to the vortex 1, respectively, and *K*<sup>1</sup> is the first order modified Bessel function. Directions of

For d-dots, there is an s-wave region between d-wave islands, and the spontaneous currents around the corners affect each other as usual supercurrent around singly quantized vortices,

 λ

and if two vortices are parallel (anti-parallel) then interaction

(24)

**4. Interaction between d-dots** 

**Figure 16.** Pairs of d-dots in a parallel (a) or a diagonal (b) positions.

is given as,

vortices are expressed by ( ) <sup>0</sup> *<sup>i</sup>*

is repulsive (attractive), respectively.

Φ

Analysis Using Two-Component Ginzburg-Landau Equations 335

**Figure 14.** Magnetic field distributions of (a) the most stable state (udud) and (b) an excited state (udud).

**Figure 15.** Free energies of (a) the most stable state (udud) and (b) an excited state (uudd). is the condensation energy of the superconductor and *Tcs* is the critical temperature of the s-wave superconductor. *E*cond

Second property is the stability of the degenerate stable states. For the square d-dots, most stable states show antiferromagnetic order of spontaneous magnetic fluxes, and we call these state udud (up down up down) (Fig. 4 (a)). There are other states, which have higher free energy. In Fig. 14 (b), one of such states is shown. In this state, spontaneous magnetic fluxes do not show antiferromagnetic order, but parallel magnetic fluxes align at the upper or lower edges. We call this state uudd (up up down down). The free energies of the udud and uudd states are shown in Fig. 15. Well below the critical temperature of the s-wave superconductor *Tcs*, free energy difference between udud (a) and uudd (b) states becomes comparable to the condensation energy of the superconductor. Therefore we can treat them as two-level systems.
