**4. Interaction between d-dots**

334 Superconductors – Materials, Properties and Applications

(udud).

superconductor.

as two-level systems.

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

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

**Figure 15.** Free energies of (a) the most stable state (udud) and (b) an excited state (uudd). is the

*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

condensation energy of the superconductor and *Tcs* is the critical temperature of the s-wave

these degenerate states using the magnetic field for asymmetric d-dots.

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.

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

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 is given as,

$$\mathbf{f}\_{12} = \frac{(\spadesuit \mathbf{0})\_1 (\spadesuit \mathbf{0})\_2}{8\pi^2 \lambda^3} K\_1 \left(\frac{r\_{12}}{\lambda}\right) \hat{\mathbf{r}}\_{12} \tag{24}$$

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 vortices are expressed by ( ) <sup>0</sup> *<sup>i</sup>* Φ and if two vortices are parallel (anti-parallel) then interaction is repulsive (attractive), respectively.

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,

mentioned above. Then, an interaction between d-dots arises through the spontaneous currents in the s-wave region. If two square d-dots are in a line (Fig. 16 (a)), nearest vortex pair should be antiparallel (e.g. right-upper flux in the left d-dot and left upper vortex in the right d-dot should be antiparallel). And then we can expect that two d-dots will have a same spontaneous magnetic flux distribution, as shown in Fig. 16 (a). Therefore when we regard a d-dot as a spin, they interact ferromagnetically. In contrast to this configuration, if two square d-dots are placed diagonally (Fig. 16 (b)), then we expect an antiferromagnetic interaction by the same argument. In Fig. 17, stable states for parallel two d-dots are shown. For short distance ((e)), the nearest magnetic fluxes disappear and the flux distributions of two d-dots are same. Increasing the distance, the nearest magnetic fluxes appear and becomes gradually large ((f)-(h)) and the states of the d-dots are still same.Therefore ferromagnetic states are always stable, as we expected, and this is independent from the distance.

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

**Figure 18.** Stable magnetic field distribution around a pair of d-dots, which are placed diagonally. The

**Figure 19.** Free energies for Ferromagnetic (F) and Antiferromagnetic states in diagonally placed two d-

However, for diagonally placed two d-dots, the magnetic field distribution is not so simple. In Fig. 18, the stable state is shown for short distance between two diagonally placed d-dots. Unlike to the expectation from the usual vortex-vortex interaction, ferromagnetic state becomes stable. This is because two adjacent half-quantum fluxes are connected and form broad single quantum flux and this can occur when two fluxes are parallel. Therefore the ferromagnetic state becomes stable. It seems that the energy of this configuration becomes lower when two half-quantum fluxes are closer to each other. This property does not appear for ordinary vortices and we call this a fusion of half-quantum vortices. In Fig. 19, distance dependence of the free energies for the ferromagnetic and antiferromagnetic states are plotted. When distance between two d-dots is short, the ferromagnetic state has much lower

dots.

states of these d-dots are same and therefore they interact ferromagnetically.

Analysis Using Two-Component Ginzburg-Landau Equations 337

**Figure 17.** Stable states for parallel two d-dots. (a)-(d) : configurations of d- and s-wave superconductors. (e)-(h): Stable magnetic field distributions.

Composite Structures of d-Wave and s-Wave Superconductors (d-Dot): Analysis Using Two-Component Ginzburg-Landau Equations 337

336 Superconductors – Materials, Properties and Applications

distance.

mentioned above. Then, an interaction between d-dots arises through the spontaneous currents in the s-wave region. If two square d-dots are in a line (Fig. 16 (a)), nearest vortex pair should be antiparallel (e.g. right-upper flux in the left d-dot and left upper vortex in the right d-dot should be antiparallel). And then we can expect that two d-dots will have a same spontaneous magnetic flux distribution, as shown in Fig. 16 (a). Therefore when we regard a d-dot as a spin, they interact ferromagnetically. In contrast to this configuration, if two square d-dots are placed diagonally (Fig. 16 (b)), then we expect an antiferromagnetic interaction by the same argument. In Fig. 17, stable states for parallel two d-dots are shown. For short distance ((e)), the nearest magnetic fluxes disappear and the flux distributions of two d-dots are same. Increasing the distance, the nearest magnetic fluxes appear and becomes gradually large ((f)-(h)) and the states of the d-dots are still same.Therefore ferromagnetic states are always stable, as we expected, and this is independent from the

**Figure 17.** Stable states for parallel two d-dots. (a)-(d) : configurations of d- and s-wave

superconductors. (e)-(h): Stable magnetic field distributions.

**Figure 18.** Stable magnetic field distribution around a pair of d-dots, which are placed diagonally. The states of these d-dots are same and therefore they interact ferromagnetically.

**Figure 19.** Free energies for Ferromagnetic (F) and Antiferromagnetic states in diagonally placed two ddots.

However, for diagonally placed two d-dots, the magnetic field distribution is not so simple. In Fig. 18, the stable state is shown for short distance between two diagonally placed d-dots. Unlike to the expectation from the usual vortex-vortex interaction, ferromagnetic state becomes stable. This is because two adjacent half-quantum fluxes are connected and form broad single quantum flux and this can occur when two fluxes are parallel. Therefore the ferromagnetic state becomes stable. It seems that the energy of this configuration becomes lower when two half-quantum fluxes are closer to each other. This property does not appear for ordinary vortices and we call this a fusion of half-quantum vortices. In Fig. 19, distance dependence of the free energies for the ferromagnetic and antiferromagnetic states are plotted. When distance between two d-dots is short, the ferromagnetic state has much lower 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 independent as shown in the insets of Fig. 19.

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

*V I* (32)

*V I* (33)

 λ

 λ

(35)

(36)

( ) { } ( ) ( ) 12 1 2 1 2

( ) { } ( ) { } ( )( ) 1 11 1 11

( ) { } 122 1 2 <sup>3</sup> 123

( ) { } 122 1 2 <sup>3</sup> 123

( ) { } ( ) ( ) 12 1 2 1 2

αα

<sup>3</sup> <sup>2</sup>

<sup>3</sup> Re

<sup>3</sup> Im

≡ Δ − −− *ds <sup>s</sup> x xx y yy ij d d ii j ji i i ii j ji i i*

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

1

2 <sup>∗</sup> Δ ≡ ΔΔ Δ *dR i iiii i i i iii*

2 <sup>∗</sup> Δ ≡ ΔΔ Δ *dI i ii i i i i i ii i*

,

 *ss s s ij d d ij i i ij i i s ij*

α

1 2

1 2

2 Re Re 3Im Im

+ Δ Δ+ Δ Δ

<sup>2</sup> 2 0

2 3Re Re Im Im

+ Δ Δ+ Δ Δ

2

*d i i*

2

*d i i*

3 2

ξ

*ss s s*

*sR <sup>s</sup>*

*sI <sup>s</sup>*

*ss <sup>s</sup>*

λ

λ

λ

λ

λ

λ

*d i*

λ

λ

<sup>2</sup> 2 0

<sup>2</sup> 3 2 0

λ

λ

*ds <sup>s</sup> xx yy*

3 4

ξ

4

2

2

ξ

1 2

*P I*

1 2

*I*

*I*

4

*d i i*

4

*d i i*

λ

λ

*ss I s*

*ss R s*

*P I*

λ

*s*

λ

λ

*s*

λ

2

ξ 1 2

=

α

≡ Δ −+ <sup>−</sup>

,

*x y P* **A** *K K I AA AA* (30)

*Q* **A** *J JA J JA* (31)

2 4 <sup>2</sup> <sup>2</sup> 2 0 <sup>0</sup>

 = = ≡ Δ + +Δ

1 2 1 2 1 2

1 2 1 2 1 2

Re Re 3Im Im

3Re Re Im Im

α α

,

*d d x y ii x y P* **A** *K I AA I* (34)

α

( ) { } ( )

Δ ≡ Δ Δ+ Δ Δ

*ij i i ij di di di di*

( )

( ) { } ( )

Δ ≡ Δ Δ+ Δ Δ

*ij i i ij di di di di*

( )

1 2 1 2 1 2

( ) { } ( ) ( ) 1 11

≡− Δ <sup>−</sup>

*ij d d ii j ji i i d i x y*

2 4 <sup>2</sup> 2Im Re 2 2Im Re

4

4

λ

λ

λ

λ

λ

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

> ( ) { } 122 1 2 <sup>3</sup> 123

> ( ) { } 122 1 2 <sup>3</sup> 123

*i i i i i si si si d ii i*

<sup>∗</sup> Δ ≡ ΔΔ Δ

4 Im

*i i i i i si si si d iii*

<sup>∗</sup> Δ ≡ ΔΔ Δ

4 Re

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

Δ ≡ Δ Δ+ Δ Δ

λ

1

α

=

*QI I* (38)

,

 λ

 λ α

*Q* **A** *J JA* (37)

*V I* (39)

*V I* (40)

 αα

2

*i i ij si si si si*

1 2 1 2 1 2

*i i ij si si si si*

Analysis Using Two-Component Ginzburg-Landau Equations 339

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 devices[27].
