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Magnetic flux structures of composite superconducting structures with d- and s-waves superconductors (d-dots)
Physica C 469 (2009) 1067–1070
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Magnetic flux structures of composite superconducting structures with d- and s-waves superconductors (d-dots) M. Kato a,e,*, T. Koyama b,e, M. Machida c,e, T. Ishida d,e a
Department of Mathematical Sciences, Osaka Prefecture University, 1-1, Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan c CCSE, Japan Atomic Energy Agency, Higashi Ueno 6-9-3, Tokyo 110-0015, Japan d Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan e JST-CREST, 5, Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan b
a r t i c l e
i n f o
Article history: Available online 30 May 2009 PACS: 74.20.De 74.78.Na 74.25.Qt
a b s t r a c t Composite superconducting structures with d- and s-wave superconductors, d-dots, can be used as two state devices. Their functions depend on structures of the spontaneous magnetic field, which appears because of the anisotropy of d-wave superconductivity. Solving two-components Ginzburg–Landau equation, we have investigated magnetic field structures for d-dots with smaller and larger holes around the corners of d-wave superconducting region. And we argued the effect of holes on the magnetic structures. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Nano-scale superconductors d-Wave superconductors Ginzburg–Landau equation
1. Introduction Recently, we proposed a new type of superconducting device, which is composed from d- and s-wave superconductors [1–15]. Especially, a small d-wave superconductor is embedded in s-wave superconducting matrix. We call it d-dot. Because of the pairing anisotropy of d-wave superconductivity, there appears spontaneous magnetic flux around an edge between d- and s-wave superconductors [16–18]. A square shaped d-wave superconductor with dx2 y2 pairing symmetry shows spontaneous half-quantum magnetic fluxes (U ¼ U0 =2 ¼ hc=4e) at every corners of the square [1,2]. These magnetic fluxes order antiferromagnetically. Every ddot that shows spontaneous magnetic flux has doubly degenerate most stable states. This is because of the broken time reversal symmetry and the magnetic flux structures of the degenerate states are completely opposite. And we do not need any external parameters for obtaining these degenerate two states. Therefore, we can use ddot as a stable two state device [11]. We established simulation method for this d-dot system, using phenomenological superconducting theory [1]. It is a two-compo-
* Corresponding author. Address: Department of Mathematical Sciences, Osaka Prefecture University, 1-1, Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan. Tel.: +81 72 254 9368; fax: +81 72 254 9916. E-mail address: [email protected] (M. Kato). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.159
nents Ginzburg–Landau (GL) theory, in which we include the d-wave and s-wave superconducting order parameters and the coupling term between them. For example, the GL free energy for d-wave superconductors are given as
2 3a 4 ln T cd =T kd jDd j2 3a 8 X h i as þakd jDs j4 þ 2 jDs j2 þ akd 2jDs j2 jDd j2
Fd ðDs ; Dd ; AÞ ¼
Z
a
h 1 1 2 2 2 þ akd v 2F jPDd j2 þ 2jPDs j2 þ D2 D þ D D s d s d 2 4 i þ Px Ds Px Dd Py Ds Py Dd þ H:c: 1 1 jh Hj2 þ ðdivAÞ2 dX; ð1Þ þ 8p 8p where Ds and Dd are s- and d-wave superconducting order paramA and Tcd is the transition temeters, respectively. And P ¼ hi r 2e c perature of the d-wave superconductor. Furthermore, we proposed new logic gates using d-dot [13,14]. They are based on the interaction of between two d-dots [11,12]. This interaction comes from the spontaneous currents, which is the origin of the spontaneous magnetic flux. The structures of these spontaneous currents are anisotropic and depend on the shape of
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M. Kato et al. / Physica C 469 (2009) 1067–1070
Fig. 1. Typical junction between s-wave and d-wave superconductors. Light grey region is an s-wave superconductor and dark grey region is a d-wave superconductors. Between two regions, there is a junction regions, where two order parameters, Ds and Dd , coexist. Solid lines show the division into the finite elements.
(a)
the s-wave superconductors. Therefore, an arrangement of several d-dots can show useful function. One of such application is the quantum dot cellular automaton idea [19,20]. Based on this idea, we successfully simulated several logic gate functions using the time-dependent two-components GL equation [13,14]. They are a transfer of information along the d-dots array and a majority-voting gate. We showed the timescale of transfer of information is an order of sub-picosecond, and therefore these devices can have a comparable ability of any other devices. We are now trying to make d-dot systems experimentally. But still we have a difficulty to establish these spontaneous magnetic structures [14]. Also for using d-dots as logic devices, it is inevitable to detect a final result efficiently. But this is somewhat difficult, because the magnetic structures are symmetric for square shaped d-dots. And it is important to investigate the magnetic flux structures and seek for effective structures d-wave superconducting regions for such detection. In this study, using two-component GL equation, we examine two kinds of d-dot configurations with holes around the corners of d-wave superconducting region and show magnetic structures for these d-dots.
(b)
1 0.8 1
0.6 0.4 0.2 0 0
0.8
0.4
y/
0.2 0.4
x/L
0.2
0.6 0.8 1
(d)
(c)
L
0.6
0
1
0.8
1 0.8 0.6 0.4 0.2 0 0
1
0.6
y/L
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0.2 0.4
x/L
0.4
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L
0.6
0.2
0.6 0.8
0.2
1
0
0 0
0.2
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1
x/L Fig. 2. Magnetic flux structure of a d-dot with smaller holes at the corners: (a) configuration, (b) spatial variations of the amplitude of d-wave order parameter, (c) spatial variations of the amplitude of s-wave order parameter and (d) magnetic field structure.
M. Kato et al. / Physica C 469 (2009) 1067–1070
(a)
(b)
(c)
(d)
1069
Fig. 3. Magnetic flux structure of a d-dot with wider holes at the corners: (a) configuration, (b) spatial variations of the amplitude of d-wave order parameter, (c) spatial variations of the amplitude of s-wave order parameter and (s) magnetic field structure.
2. Method
3. Numerical results
We start from the two-component GL equation. But unlike to previous model, we consider junction region between d- and swave superconducting regions, as shown in Fig. 1. And we only consider the coupling between two order parameters, Ds and Dd in this region. In the s-wave and d-wave regions, we only include one of Ds and Dd in the free energies. The free energy for the junction region is given as
In the following, we show magnetic flux structures for several d-dot configurations. First, we examine how existence of the small hole at the corner affect the spontaneous magnetic structures. In Fig. 2a, we show the configuration of the d- and s-wave superconducting regions. At the four corners, there are vacuum regions in the junction and s-wave regions. We have solved the two-component GL equations for this configuration. The distribution of the dwave (s-wave) order parameter is shown in Fig. 2b (Fig. 2c), respectively. These order parameters are well isolated and mixture only in the junction region. The distribution of spontaneous magnetic field is shown in Fig. 2d. From this figure, we can say the magnetic field is not much affected by the holes, but the distribution becomes rather broad. This is because the spontaneous current, which causes the magnetic flux, flows around the holes. Next we examine the longer and wider hole, which is shown in Fig. 3a. The order parameter structures are shown in Figs. 3b and c. Distribution of the magnetic field is shown in Fig. 3d. In this case, magnetic field appears around corners of the d-wave superconductor. The spontaneous current flows along the boundary between the vacuum and the d-wave superconductor. This is because the phase of the d-wave order parameter is fixed at the regions where s-wave superconductor is attached through the junction region. If there is no variation of the s-wave superconductor around the corner, the fixed phases are different at both sides of a corner, because of the anisotropy of the d-wave order parameter. Therefore, the phase variation occurs inside of the d-wave superconductor. This
Z h 3b h ad i2 as i2 jkd j þ b j Ds j 2 þ jkd j jDd j2 þ 2 3b 2b X 1 þbjkd j 2jDs j2 jDd j2 þ D2 D2 þ D2s D2 d 2 s d 1 n þ v 2F jPDd j2 þ 2jPDs j2 þ Px Ds Px Dd 4 oi
F M ðDs ; Dd ; AÞ ¼
Py Ds Py Dd þ H:c: þ
1 1 jh Hj2 þ ðdivAÞ2 dX: 8p 8p
ð2Þ
In order to minimize the GL free energies, we use the finite element method. In the finite element method the regions are divided into the finite elements. Typical finite element division is shown in Fig. 1. In this figure, the bright grey and dark grey regions are swave and d-wave superconductors, respectively. Between them, there is a junction region. And because the spatial variation of order parameters is large, the sizes of the elements are small in this region.
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M. Kato et al. / Physica C 469 (2009) 1067–1070
magnetic field distribution does not change the fundamental properties of d-dot system, such as, double degenerate ground states. But it is not good for the detection of the state of the d-dot system. 4. Summary We examined two composite structures of d- and s-wave superconductors for realistic constructions of d-dot systems. The existence of the smaller holes at the corner junctions does not affect the spontaneous magnetic structures. But the larger holes affect the flow pattern of the spontaneous current. Especially, the current flows inside of the d-wave superconductor. In order to construct the d-dot system and to consider the structures for the detection, this effect is important. Acknowledgment This work is partially supported by Grant-in-Aid for the Scientific Research from the Japan Society for the Promotion of Science. References [1] M. Kato, M. Ako, M. Machida, T. Koyama, T. Ishida, Physica C 412–414 (2004) 352.
[2] M. Ako, M. Machida, T. Koyama, T. Ishida, M. Kato, Physica C 412–414 (2004) 544. [3] T. Koyama, M. Machida, M. Kato, T. Ishida, Physica C 412–414 (2004) 358. [4] M. Machida, T. Koyama, M. Kato, T. Ishida, Physica C 412–414 (2004) 367. [5] M. Ako, M. Machida, T. Koyama, T. Ishida, M. Kato, Physica C 426–431 (2005) 122. [6] M. Hirayama, M. Machida, T. Koyama, T. Ishida, M. Kato, Physica C 426–431 (2005) 127. [7] T. Koyama, M. Machida, M. Kato, T. Ishida, Physica C 426–431 (2005) 1561. [8] M. Machida, T. Koyama, M. Kato, T. Ishida, Physica C 426–431 (2005) 1566. [9] M. Kato, M. Hirayama, M. Machida, T. Koyama, T. Ishida, AIP Conf. Proc. 850 (2006) 781. [10] T. Koyama, M. Machida, M. Kato, T. Ishida, AIP Conf. Proc. 850 (2006) 779. [11] M. Kato, M. Hirayama, H. Suematsu, M. Machida, T. Koyama, T. Ishida, J. Korean Phys. Soc. 48 (2006) 1074. [12] M. Hirayama, M. Kato, M. Machida, T. Koyama, T. Ishida, Physica C 445–448 (2006) 264. [13] S. Nakajima, M. Kato, T. Koyama, M. Machida, T. Ishida, F. Nori, Physica C 468 (2008) 769. [14] S. Nakajima, M. Kato, M. Machida, T. Koyama, T. Ishida, F. Nori, Physica C 468 (2008) 1910. [15] T. Ishida, M. Fujii, T. Abe, S. Miki, S. Kawamata, K. Satoh, T. Yotsuya, M. Kato, M. Machida, T. Koyama, T. Terashima, S. Tsukui, M. Adachi, Physica C 437–438 (2006) 104. [16] C.C. Tsuei, J.R. Kirtley, Rev. Mod. Phys. 72 (2000) 969. [17] D.J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515. [18] A. Hilgenkamp, Ariando, H.-J.H. Smilde, D.H.A. Blank, G. Rijnders, H. Rogalla, J.R. Kirtley, C.C. Tsuei, Nature 422 (2003) 50. [19] C.S. Lent, P.D. Tougaw, W. Porod, J. Appl. Phys. 74 (1993) 3558. [20] C.S. Lent, P.D. Taugaw, W. Porod, G.H. Bernstein, Nanotechnology 4 (1993) 49.
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Magnetic flux structures of composite superconducting structures with d- and s-waves superconductors (d-dots) M. Kato a,e,*, T. Koyama b,e, M. Machida c,e, T. Ishida d,e a
Department of Mathematical Sciences, Osaka Prefecture University, 1-1, Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan c CCSE, Japan Atomic Energy Agency, Higashi Ueno 6-9-3, Tokyo 110-0015, Japan d Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan e JST-CREST, 5, Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan b
a r t i c l e
i n f o
Article history: Available online 30 May 2009 PACS: 74.20.De 74.78.Na 74.25.Qt
a b s t r a c t Composite superconducting structures with d- and s-wave superconductors, d-dots, can be used as two state devices. Their functions depend on structures of the spontaneous magnetic field, which appears because of the anisotropy of d-wave superconductivity. Solving two-components Ginzburg–Landau equation, we have investigated magnetic field structures for d-dots with smaller and larger holes around the corners of d-wave superconducting region. And we argued the effect of holes on the magnetic structures. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Nano-scale superconductors d-Wave superconductors Ginzburg–Landau equation
1. Introduction Recently, we proposed a new type of superconducting device, which is composed from d- and s-wave superconductors [1–15]. Especially, a small d-wave superconductor is embedded in s-wave superconducting matrix. We call it d-dot. Because of the pairing anisotropy of d-wave superconductivity, there appears spontaneous magnetic flux around an edge between d- and s-wave superconductors [16–18]. A square shaped d-wave superconductor with dx2 y2 pairing symmetry shows spontaneous half-quantum magnetic fluxes (U ¼ U0 =2 ¼ hc=4e) at every corners of the square [1,2]. These magnetic fluxes order antiferromagnetically. Every ddot that shows spontaneous magnetic flux has doubly degenerate most stable states. This is because of the broken time reversal symmetry and the magnetic flux structures of the degenerate states are completely opposite. And we do not need any external parameters for obtaining these degenerate two states. Therefore, we can use ddot as a stable two state device [11]. We established simulation method for this d-dot system, using phenomenological superconducting theory [1]. It is a two-compo-
* Corresponding author. Address: Department of Mathematical Sciences, Osaka Prefecture University, 1-1, Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan. Tel.: +81 72 254 9368; fax: +81 72 254 9916. E-mail address: [email protected] (M. Kato). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.05.159
nents Ginzburg–Landau (GL) theory, in which we include the d-wave and s-wave superconducting order parameters and the coupling term between them. For example, the GL free energy for d-wave superconductors are given as
2 3a 4 ln T cd =T kd jDd j2 3a 8 X h i as þakd jDs j4 þ 2 jDs j2 þ akd 2jDs j2 jDd j2
Fd ðDs ; Dd ; AÞ ¼
Z
a
h 1 1 2 2 2 þ akd v 2F jPDd j2 þ 2jPDs j2 þ D2 D þ D D s d s d 2 4 i þ Px Ds Px Dd Py Ds Py Dd þ H:c: 1 1 jh Hj2 þ ðdivAÞ2 dX; ð1Þ þ 8p 8p where Ds and Dd are s- and d-wave superconducting order paramA and Tcd is the transition temeters, respectively. And P ¼ hi r 2e c perature of the d-wave superconductor. Furthermore, we proposed new logic gates using d-dot [13,14]. They are based on the interaction of between two d-dots [11,12]. This interaction comes from the spontaneous currents, which is the origin of the spontaneous magnetic flux. The structures of these spontaneous currents are anisotropic and depend on the shape of
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M. Kato et al. / Physica C 469 (2009) 1067–1070
Fig. 1. Typical junction between s-wave and d-wave superconductors. Light grey region is an s-wave superconductor and dark grey region is a d-wave superconductors. Between two regions, there is a junction regions, where two order parameters, Ds and Dd , coexist. Solid lines show the division into the finite elements.
(a)
the s-wave superconductors. Therefore, an arrangement of several d-dots can show useful function. One of such application is the quantum dot cellular automaton idea [19,20]. Based on this idea, we successfully simulated several logic gate functions using the time-dependent two-components GL equation [13,14]. They are a transfer of information along the d-dots array and a majority-voting gate. We showed the timescale of transfer of information is an order of sub-picosecond, and therefore these devices can have a comparable ability of any other devices. We are now trying to make d-dot systems experimentally. But still we have a difficulty to establish these spontaneous magnetic structures [14]. Also for using d-dots as logic devices, it is inevitable to detect a final result efficiently. But this is somewhat difficult, because the magnetic structures are symmetric for square shaped d-dots. And it is important to investigate the magnetic flux structures and seek for effective structures d-wave superconducting regions for such detection. In this study, using two-component GL equation, we examine two kinds of d-dot configurations with holes around the corners of d-wave superconducting region and show magnetic structures for these d-dots.
(b)
1 0.8 1
0.6 0.4 0.2 0 0
0.8
0.4
y/
0.2 0.4
x/L
0.2
0.6 0.8 1
(d)
(c)
L
0.6
0
1
0.8
1 0.8 0.6 0.4 0.2 0 0
1
0.6
y/L
0.8
0.4
0.2 0.4
x/L
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y/
L
0.6
0.2
0.6 0.8
0.2
1
0
0 0
0.2
0.4
0.6
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1
x/L Fig. 2. Magnetic flux structure of a d-dot with smaller holes at the corners: (a) configuration, (b) spatial variations of the amplitude of d-wave order parameter, (c) spatial variations of the amplitude of s-wave order parameter and (d) magnetic field structure.
M. Kato et al. / Physica C 469 (2009) 1067–1070
(a)
(b)
(c)
(d)
1069
Fig. 3. Magnetic flux structure of a d-dot with wider holes at the corners: (a) configuration, (b) spatial variations of the amplitude of d-wave order parameter, (c) spatial variations of the amplitude of s-wave order parameter and (s) magnetic field structure.
2. Method
3. Numerical results
We start from the two-component GL equation. But unlike to previous model, we consider junction region between d- and swave superconducting regions, as shown in Fig. 1. And we only consider the coupling between two order parameters, Ds and Dd in this region. In the s-wave and d-wave regions, we only include one of Ds and Dd in the free energies. The free energy for the junction region is given as
In the following, we show magnetic flux structures for several d-dot configurations. First, we examine how existence of the small hole at the corner affect the spontaneous magnetic structures. In Fig. 2a, we show the configuration of the d- and s-wave superconducting regions. At the four corners, there are vacuum regions in the junction and s-wave regions. We have solved the two-component GL equations for this configuration. The distribution of the dwave (s-wave) order parameter is shown in Fig. 2b (Fig. 2c), respectively. These order parameters are well isolated and mixture only in the junction region. The distribution of spontaneous magnetic field is shown in Fig. 2d. From this figure, we can say the magnetic field is not much affected by the holes, but the distribution becomes rather broad. This is because the spontaneous current, which causes the magnetic flux, flows around the holes. Next we examine the longer and wider hole, which is shown in Fig. 3a. The order parameter structures are shown in Figs. 3b and c. Distribution of the magnetic field is shown in Fig. 3d. In this case, magnetic field appears around corners of the d-wave superconductor. The spontaneous current flows along the boundary between the vacuum and the d-wave superconductor. This is because the phase of the d-wave order parameter is fixed at the regions where s-wave superconductor is attached through the junction region. If there is no variation of the s-wave superconductor around the corner, the fixed phases are different at both sides of a corner, because of the anisotropy of the d-wave order parameter. Therefore, the phase variation occurs inside of the d-wave superconductor. This
Z h 3b h ad i2 as i2 jkd j þ b j Ds j 2 þ jkd j jDd j2 þ 2 3b 2b X 1 þbjkd j 2jDs j2 jDd j2 þ D2 D2 þ D2s D2 d 2 s d 1 n þ v 2F jPDd j2 þ 2jPDs j2 þ Px Ds Px Dd 4 oi
F M ðDs ; Dd ; AÞ ¼
Py Ds Py Dd þ H:c: þ
1 1 jh Hj2 þ ðdivAÞ2 dX: 8p 8p
ð2Þ
In order to minimize the GL free energies, we use the finite element method. In the finite element method the regions are divided into the finite elements. Typical finite element division is shown in Fig. 1. In this figure, the bright grey and dark grey regions are swave and d-wave superconductors, respectively. Between them, there is a junction region. And because the spatial variation of order parameters is large, the sizes of the elements are small in this region.
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M. Kato et al. / Physica C 469 (2009) 1067–1070
magnetic field distribution does not change the fundamental properties of d-dot system, such as, double degenerate ground states. But it is not good for the detection of the state of the d-dot system. 4. Summary We examined two composite structures of d- and s-wave superconductors for realistic constructions of d-dot systems. The existence of the smaller holes at the corner junctions does not affect the spontaneous magnetic structures. But the larger holes affect the flow pattern of the spontaneous current. Especially, the current flows inside of the d-wave superconductor. In order to construct the d-dot system and to consider the structures for the detection, this effect is important. Acknowledgment This work is partially supported by Grant-in-Aid for the Scientific Research from the Japan Society for the Promotion of Science. References [1] M. Kato, M. Ako, M. Machida, T. Koyama, T. Ishida, Physica C 412–414 (2004) 352.
[2] M. Ako, M. Machida, T. Koyama, T. Ishida, M. Kato, Physica C 412–414 (2004) 544. [3] T. Koyama, M. Machida, M. Kato, T. Ishida, Physica C 412–414 (2004) 358. [4] M. Machida, T. Koyama, M. Kato, T. Ishida, Physica C 412–414 (2004) 367. [5] M. Ako, M. Machida, T. Koyama, T. Ishida, M. Kato, Physica C 426–431 (2005) 122. [6] M. Hirayama, M. Machida, T. Koyama, T. Ishida, M. Kato, Physica C 426–431 (2005) 127. [7] T. Koyama, M. Machida, M. Kato, T. Ishida, Physica C 426–431 (2005) 1561. [8] M. Machida, T. Koyama, M. Kato, T. Ishida, Physica C 426–431 (2005) 1566. [9] M. Kato, M. Hirayama, M. Machida, T. Koyama, T. Ishida, AIP Conf. Proc. 850 (2006) 781. [10] T. Koyama, M. Machida, M. Kato, T. Ishida, AIP Conf. Proc. 850 (2006) 779. [11] M. Kato, M. Hirayama, H. Suematsu, M. Machida, T. Koyama, T. Ishida, J. Korean Phys. Soc. 48 (2006) 1074. [12] M. Hirayama, M. Kato, M. Machida, T. Koyama, T. Ishida, Physica C 445–448 (2006) 264. [13] S. Nakajima, M. Kato, T. Koyama, M. Machida, T. Ishida, F. Nori, Physica C 468 (2008) 769. [14] S. Nakajima, M. Kato, M. Machida, T. Koyama, T. Ishida, F. Nori, Physica C 468 (2008) 1910. [15] T. Ishida, M. Fujii, T. Abe, S. Miki, S. Kawamata, K. Satoh, T. Yotsuya, M. Kato, M. Machida, T. Koyama, T. Terashima, S. Tsukui, M. Adachi, Physica C 437–438 (2006) 104. [16] C.C. Tsuei, J.R. Kirtley, Rev. Mod. Phys. 72 (2000) 969. [17] D.J. Van Harlingen, Rev. Mod. Phys. 67 (1995) 515. [18] A. Hilgenkamp, Ariando, H.-J.H. Smilde, D.H.A. Blank, G. Rijnders, H. Rogalla, J.R. Kirtley, C.C. Tsuei, Nature 422 (2003) 50. [19] C.S. Lent, P.D. Tougaw, W. Porod, J. Appl. Phys. 74 (1993) 3558. [20] C.S. Lent, P.D. Taugaw, W. Porod, G.H. Bernstein, Nanotechnology 4 (1993) 49.