- Email: [email protected]
Theory of finite temperature Josephson transport through a ferromagnetic insulator
Available online at www.sciencedirect.com
Physics Procedia 27 (2012) 308 – 311
ISS2011
Theory of finite temperature Josephson transport through a ferromagnetic insulator Shuhei Nakamuraa , Satofumi Soumaa , Matsuto Ogawaa , Shiro Kawabatab,c,∗ a Department b Nanosystem
of Electrical and Electronic Engineering, Kobe University, 1-1, Rokko-dai, Nada-ku, Kobe, 657-8501, Japan Research Institute (NRI), National Institute of Advanced Industrial Science and Technology (AIST), Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan c CREST, Japan Science and Technology Corporation (JST), Kawaguchi, Saitama 332-0012, Japan
Abstract We predict an anomalous 0-π transition in Josephson junctions with a ferromagnetic-insulator (FI) barrier. Previously it was found that the ground state of such junctions alternates between 0 and π states when the thickness of FI is increasing by a single atomic layer. By solving the Bogoliubov de-Gennes equation and using the Furusaki-Tsukada formula, we show that similar 0-π transition can be also induced by increasing temperature T . Therefore the existence of π state can be experimentally confirmed by simply observing the nonmonotonic T dependence of the Josephson critical current.
© by Elsevier ElsevierLtd. B.V. Selection and/or peer-review under responsibility of ISS Program Committee c 2012 2011 Published by Keywords: Josephson junction ; Ferromagnetic insulator ; Tight binding model ; Spintronics ; Quantum computer
1. Introduction A superconducting phase difference of π can be arranged between two superconductors (Ss) when separating them by a suitably chosen ferromagnetic metals (FMs) [1, 2]. As a result, in the π-state, the Josephson current phase relation in such S/FM/S junctions is given by I J = IC sin φ, where IC < 0 is the Josephson critical current and φ is the phase difference between two Ss. Transition between the π state (negative IC ) and the 0 state (positive IC ) has been revealed in experiments through oscillations of IC with varying thickness of the FM [3] or with varying temperature T [4]. Currently the π junction is of considerable interest as an element complementary to the usual Josephson junction in the development of functional nano-structures including superconducting classical and quantum electronics [5, 6, 7, 8, 9, 10, 11]. Recently, Josephson transport through ferromagnetic insulators (FIs) has been investigated theoretically [12, 13, 14] and experimentally [15]. In order to study the transport properties of such systems by taking into account the band structure of FIs explicitly, an efficient numerical method to calculate the Josephson current based on the Bogoliubov de-Gennes (BdG) equation has been developed by one of the authors [12]. By use of this method, we have predicted the formation of the π-state in such systems. Moreover we have found that an atomic scale 0-π transition is induced by increasing the thickness of the FI layer ∗ Corresponding author. TEL: +81-29-861-5390; fax: +81-29-861-5375. E-mail address: [email protected]
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of ISS Program Committee doi:10.1016/j.phpro.2012.03.472
309
Shuhei Nakamura et al. / Physics Procedia 27 (2012) 308 – 311
one by one [13, 14]. Up to now, however, only the Josephson current at the zero temperature limit has been explored. In this paper we will investigate the f inite-temperature Josephson transport and show that the 0-π phase transition can be also realized by increasing the temperature. Therefore the existence of π state can be confirmed experimentally by simply measuring the temperature dependence of IC . 2. Model In this section, we show the numerical method to calculate the Josephson current. Let us consider an S/FI/S junction in a one-dimensional tight-binding model as shown in Fig. 1. The thickness of a FI layer is defined by LF and j points to a lattice site. The electronic states in an s-wave superconductor are described by the mean-field BCS Hamiltonian, H = −t (Ψ†j+1,σ Ψ j,σ + Ψ†j,σ Ψ j+1,σ ) + (2t − μ)Ψ†j,σ Ψ j,σ + (ΔΨ†j↑ Ψ†j↓ + Δ∗ Ψ j↓ Ψ j↑ ). (1) j,σ
j,σ
j
Here Ψ†j,σ (Ψ j,σ ) is the creation (annihilation) operator of an electron at j with spin σ = (↑ or ↓) and μ is the Fermi energy. The hopping energy t is considered between nearest neighbor sites and Δ is the amplitude of s-wave pair potential. The Hamiltonian can be expressed by a 4×4 matrix form as ⎤ ⎤⎡ ⎡ 0 0 Δ( j, j ) ⎥⎥ ⎢⎢⎢Ψ j ↑ ⎥⎥⎥ ⎢⎢⎢h0 ( j, j ) ⎢ ⎥ ⎥⎥⎥ ⎢⎢⎢Ψ j ↓ ⎥⎥⎥⎥ ⎢⎢⎢ 0 h0 ( j, j ) −Δ( j, j ) 0 ⎥⎥⎥ ⎢⎢⎢ † ⎥⎥⎥ , (2) H= [Ψ†j↑ Ψ†j↓ Ψ j↑ Ψ j↓ ] ⎢⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢Ψ j ↑ ⎥⎥⎥ −Δ( j, j ) −h0 ( j, j ) 0 ⎢⎢⎣ 0 ⎢ ⎦ ⎥ jj ⎣ ⎦ Δ( j, j ) 0 0 −h0 ( j, j ) Ψ†j ↓ where h0 ( j, j ) = −t(δ j, j +1 +δ j, j −1 )+(2t−μ)δ j j and Δ( j, j ) = Δeiϕ δ j j . The Hamiltonian can be diagonalized by the Bogoliubov transformation. In what follows, we focus on the subspace for spin-↑ electron and spin-↓ hole. In superconductors, the wave function is given by
ˆ L u αeik j + v βe−ik∗ j + u Ae−ik j + v Beik∗ j , ΨL ( j) = Φ (3) v u v u
ˆ R u Ceik j + v De−ik∗ j . ΨR ( j) = Φ (4) v u Here ν =L (R) indicates a superconductor in the √ left-(right-)hand side, Φν is the phase factor of a superϕν ϕν conductor, Φν =diag(ei 2 , e−i 2 ), u(v) = [(1 + (−) E 2 − Δ2 /E)/2]1/2 , where the energy E is measured from μ. The complex wave number k for a quasiparticle is given k = π/2 + i cosh−1 1 + (Δ2 − E 2 )/4t2 . On the other hands, the Hamiltonian for a FI barrier can be described by, ⎤ ⎡⎢Ψ j ↑ ⎤⎥ ⎡ 0 0 0 ⎥ ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢h ( j, j ) − V ⎥⎥⎥ ⎢⎢⎢Ψ j ↓ ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 h ( j, j ) + V 0 0 † † ⎥ ⎢ ⎢ ⎥⎥⎥ ⎢⎢⎢ † ⎥⎥⎥⎥ H = [Ψ j↑ Ψ j↓ Ψ j↑ Ψ j↓ ] ⎢⎢⎢ 0 0 −h ( j, j ) + V 0 ⎢⎣ ⎥⎥⎦ ⎢⎢⎢Ψ j ↑ ⎥⎥⎥ ⎢ jj 0 0 0 −h ( j, j ) − V ⎣Ψ†j ↓ ⎦ (5)
Fig. 1. The Josephson junction with a ferromagnetic-insulator (FI) barrier on the one-dimensional tight-binding lattice.
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Shuhei Nakamura et al. / Physics Procedia 27 (2012) 308 – 311
Fig. 2. The temperature T dependence of Josephson critical current IC for the junction with (a) odd LF (LF = 3) and (b) even LF (LF = 4). g is the band gap of FI. TC is the superconducting transition temperature. We set Δ = 0.01t.
in the matrix form [13]. Here h ( j, j ) = −t(δ j, j +1 + δ j, j −1 ), V = 2t + g is the exchange splitting of FI with g being the gap between spin up & down bands. The wave function of a bound state in FI is given by
fR f 0 βj 0 −β j ΨFI ( j) = (−1) j e−β j + L (−1) j eβ j + e + e , (6) gR gL 0 0 where β = cosh−1 (1 + E/2t + g/4t) is the imaginary part of the quasiparticle wave-number. By applying boundary conditions, ΨL (0) = ΨFI (0), ΨL (1) = ΨFI (1), ΨR (LF ) = ΨFI (LF ) and ΨR (LF + 1) = ΨFI (LF + 1), we can obtain the Andreev reflection coefficients rhe and reh analytically. The Josephson current can be calculated from the Furusaki-Tsukada formula [16, 17], IJ =
Δ e rhe − reh , kB T Ωn Ω
(7)
n
where T is the temperature and Ωn = ω2n + Δ2 with ωn = (2n + 1)πkB T being the Matsubara frequency. As long as we consider the case with g > Δ, the Josephson current phase relation is given by I J = IC sin φ. Thus we can compute IC from the relation IC = I J (φ = π/2). 3. Results In this section, we show the numerical results for the Josephson current I J in the finite temperature regime. Fig. 2 shows the temperature T dependence of the Josephson critical current IC for the cases that LF is (a) odd (LF = 3) and (b) even (LF = 4). In the zero-temperature (T TC ) limit, it is clear from Fig. 2 that IC is negative (positive) for odd (even) L. This observation is consistent with previous theoretical results, i. e., the atomic scale 0-π transition [13]. Importantly we have found the π-0 (0-π) transition by increasing the temperature T for the junction with odd (even) LF . This thermally induced 0-π transition becomes more prominent in the small-barrier case (g t). Therefore we can experimentally confirm the existence of the π state in S/FI/S junctions by simply observing the nonmonotonic T dependence of |IC |. The physical mechanism behind this anomalous phenomena will be given elsewhere. On the other hand, in the high-barrier limit (g t), we found that IC shows simple Ambegaokar-Baratoff type monotonic behavior and does not change its sign. This result indicates that the π (0) state is quite stable against the temperature variation. In an actual FI material, e.g., La2 BaCuO5 [18, 19], g/t ≈ 11 1 [13]. Thus we can expect a robust π state formation in junctions with these FI barriers. Such a robust property is very important for future electronic device applications.
Shuhei Nakamura et al. / Physics Procedia 27 (2012) 308 – 311
4. Conclusion To summarize, we have theoretically investigated the finite Temperature Josephson transport in a S/FI/S junction by solving the B-dG equation. The 0-π transition is induced by increasing the temperature T in the low barrier cases (g < t). This indicates that the existence of the π state can be experimentally confirmed by simply measuring the nonmonotonic T dependence of IC . In the high-barrier limit (g t), on the other hand, we found that the atomic-scale 0-π transition is robust against the T variation. Therefore, such FIbased Josephson junctions will be a promising candidate for a coherent and robust superconducting quantum bit as well as a classical superconducting digital circuit. 5. Acknowledgements We would like to thank Y. Asano for useful discussion. This work was supported by JST-CREST, the ”Topological Quantum Phenomena” (No. 22103002) KAKENHI on Innovative Areas, and a Grant-in-Aid for Scientific Research (No. 22710096) from MEXT of Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A. I. Buzdin, Rev. Mod. Phys. 77 (2005) 935. A. A. Golubov, M. Y. Kupriyanov, E. Il’ichev, Rev. Mod. Phys. 76 (2004) 411. T. Kontos, M. Aprili, J. Lesueur, F. Genˆet, B. Stephanidis, R. Boursier, Phys. Rev. Lett. 89 (2002) 137007. V. V. Ryazanov, V. A. Oboznov, A. Y. Rusanov, A. V. Veretennikov, A. A. Golubov, J. Aarts, Phys. Rev. Lett. 86 (2001) 2427. L. B. Ioffe, V. B. Geshkenbein, M. V. Feigel’man, A. L. Fauch´ere, G. Blatter, Nature 398 (1999) 679. G. Blatter, V. B. Geshkenbein, L. B. Ioffe, Phys. Rev. B 63 (2001) 174511. S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, Physica C 437-438 (2006) 136. S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, A. A. Golubov, Phys. Rev. B 74 (2006) 180502(R). T. Ortlepp, Ariando, O. Mielke, C. J. M. Verwijs, K. F. K. Foo, H. Rogalla, F. H. Uhlmann, H. Hilgenkamp, Science 312 (2006) 1495. S. Kawabata, A. A. Golubov, Physica E 40 (2007) 386. A. K. Feofanov, V. A. Oboznov, V. V. Bolginov, J. Lisenfeld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko, M. Khabipov, D. Balashov, A. B. Zorin, P. N. Dmitriev, V. P. Koshelets, A. V. Ustinov, Nature Phys. 6 (2010) 593. S. Kawabata, Y. Asano, Int. J. Mod. Phys. B 23 (2009) 4329. S. Kawabata, Y. Asano, Y. Tanaka, A. A. Golubov, S. Kashiwaya, Phys. Rev. Lett. 104 (2010) 117002. S. Kawabata, Y. Asano, Low. Temp. Phys. 36 (2010) 915. K. Senapati, M. G. Blamire, Z. H. Barber, Nature Mat. 10 (2011) 849. A. Furusaki, M. Tsukada, Solid State Commun. 78 (1991) 299. A. Furusaki, M. Tsukada, Phys. Rev. B 43 (1991) 10164. F. Mizuno, H. Masuda, I. Hirabayashi, S. Tanaka, M. Hasegawa, U. Mizutani, Nature 345 (1990) 788. V. Eyert, K. H. H¨oc, and P. S. Riseborough, Europhys. Lett. 31 (1995) 385.
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Physics Procedia 27 (2012) 308 – 311
ISS2011
Theory of finite temperature Josephson transport through a ferromagnetic insulator Shuhei Nakamuraa , Satofumi Soumaa , Matsuto Ogawaa , Shiro Kawabatab,c,∗ a Department b Nanosystem
of Electrical and Electronic Engineering, Kobe University, 1-1, Rokko-dai, Nada-ku, Kobe, 657-8501, Japan Research Institute (NRI), National Institute of Advanced Industrial Science and Technology (AIST), Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan c CREST, Japan Science and Technology Corporation (JST), Kawaguchi, Saitama 332-0012, Japan
Abstract We predict an anomalous 0-π transition in Josephson junctions with a ferromagnetic-insulator (FI) barrier. Previously it was found that the ground state of such junctions alternates between 0 and π states when the thickness of FI is increasing by a single atomic layer. By solving the Bogoliubov de-Gennes equation and using the Furusaki-Tsukada formula, we show that similar 0-π transition can be also induced by increasing temperature T . Therefore the existence of π state can be experimentally confirmed by simply observing the nonmonotonic T dependence of the Josephson critical current.
© by Elsevier ElsevierLtd. B.V. Selection and/or peer-review under responsibility of ISS Program Committee c 2012 2011 Published by Keywords: Josephson junction ; Ferromagnetic insulator ; Tight binding model ; Spintronics ; Quantum computer
1. Introduction A superconducting phase difference of π can be arranged between two superconductors (Ss) when separating them by a suitably chosen ferromagnetic metals (FMs) [1, 2]. As a result, in the π-state, the Josephson current phase relation in such S/FM/S junctions is given by I J = IC sin φ, where IC < 0 is the Josephson critical current and φ is the phase difference between two Ss. Transition between the π state (negative IC ) and the 0 state (positive IC ) has been revealed in experiments through oscillations of IC with varying thickness of the FM [3] or with varying temperature T [4]. Currently the π junction is of considerable interest as an element complementary to the usual Josephson junction in the development of functional nano-structures including superconducting classical and quantum electronics [5, 6, 7, 8, 9, 10, 11]. Recently, Josephson transport through ferromagnetic insulators (FIs) has been investigated theoretically [12, 13, 14] and experimentally [15]. In order to study the transport properties of such systems by taking into account the band structure of FIs explicitly, an efficient numerical method to calculate the Josephson current based on the Bogoliubov de-Gennes (BdG) equation has been developed by one of the authors [12]. By use of this method, we have predicted the formation of the π-state in such systems. Moreover we have found that an atomic scale 0-π transition is induced by increasing the thickness of the FI layer ∗ Corresponding author. TEL: +81-29-861-5390; fax: +81-29-861-5375. E-mail address: [email protected]
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of ISS Program Committee doi:10.1016/j.phpro.2012.03.472
309
Shuhei Nakamura et al. / Physics Procedia 27 (2012) 308 – 311
one by one [13, 14]. Up to now, however, only the Josephson current at the zero temperature limit has been explored. In this paper we will investigate the f inite-temperature Josephson transport and show that the 0-π phase transition can be also realized by increasing the temperature. Therefore the existence of π state can be confirmed experimentally by simply measuring the temperature dependence of IC . 2. Model In this section, we show the numerical method to calculate the Josephson current. Let us consider an S/FI/S junction in a one-dimensional tight-binding model as shown in Fig. 1. The thickness of a FI layer is defined by LF and j points to a lattice site. The electronic states in an s-wave superconductor are described by the mean-field BCS Hamiltonian, H = −t (Ψ†j+1,σ Ψ j,σ + Ψ†j,σ Ψ j+1,σ ) + (2t − μ)Ψ†j,σ Ψ j,σ + (ΔΨ†j↑ Ψ†j↓ + Δ∗ Ψ j↓ Ψ j↑ ). (1) j,σ
j,σ
j
Here Ψ†j,σ (Ψ j,σ ) is the creation (annihilation) operator of an electron at j with spin σ = (↑ or ↓) and μ is the Fermi energy. The hopping energy t is considered between nearest neighbor sites and Δ is the amplitude of s-wave pair potential. The Hamiltonian can be expressed by a 4×4 matrix form as ⎤ ⎤⎡ ⎡ 0 0 Δ( j, j ) ⎥⎥ ⎢⎢⎢Ψ j ↑ ⎥⎥⎥ ⎢⎢⎢h0 ( j, j ) ⎢ ⎥ ⎥⎥⎥ ⎢⎢⎢Ψ j ↓ ⎥⎥⎥⎥ ⎢⎢⎢ 0 h0 ( j, j ) −Δ( j, j ) 0 ⎥⎥⎥ ⎢⎢⎢ † ⎥⎥⎥ , (2) H= [Ψ†j↑ Ψ†j↓ Ψ j↑ Ψ j↓ ] ⎢⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢Ψ j ↑ ⎥⎥⎥ −Δ( j, j ) −h0 ( j, j ) 0 ⎢⎢⎣ 0 ⎢ ⎦ ⎥ jj ⎣ ⎦ Δ( j, j ) 0 0 −h0 ( j, j ) Ψ†j ↓ where h0 ( j, j ) = −t(δ j, j +1 +δ j, j −1 )+(2t−μ)δ j j and Δ( j, j ) = Δeiϕ δ j j . The Hamiltonian can be diagonalized by the Bogoliubov transformation. In what follows, we focus on the subspace for spin-↑ electron and spin-↓ hole. In superconductors, the wave function is given by
ˆ L u αeik j + v βe−ik∗ j + u Ae−ik j + v Beik∗ j , ΨL ( j) = Φ (3) v u v u
ˆ R u Ceik j + v De−ik∗ j . ΨR ( j) = Φ (4) v u Here ν =L (R) indicates a superconductor in the √ left-(right-)hand side, Φν is the phase factor of a superϕν ϕν conductor, Φν =diag(ei 2 , e−i 2 ), u(v) = [(1 + (−) E 2 − Δ2 /E)/2]1/2 , where the energy E is measured from μ. The complex wave number k for a quasiparticle is given k = π/2 + i cosh−1 1 + (Δ2 − E 2 )/4t2 . On the other hands, the Hamiltonian for a FI barrier can be described by, ⎤ ⎡⎢Ψ j ↑ ⎤⎥ ⎡ 0 0 0 ⎥ ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢h ( j, j ) − V ⎥⎥⎥ ⎢⎢⎢Ψ j ↓ ⎥⎥⎥⎥⎥ ⎢⎢⎢ 0 h ( j, j ) + V 0 0 † † ⎥ ⎢ ⎢ ⎥⎥⎥ ⎢⎢⎢ † ⎥⎥⎥⎥ H = [Ψ j↑ Ψ j↓ Ψ j↑ Ψ j↓ ] ⎢⎢⎢ 0 0 −h ( j, j ) + V 0 ⎢⎣ ⎥⎥⎦ ⎢⎢⎢Ψ j ↑ ⎥⎥⎥ ⎢ jj 0 0 0 −h ( j, j ) − V ⎣Ψ†j ↓ ⎦ (5)
Fig. 1. The Josephson junction with a ferromagnetic-insulator (FI) barrier on the one-dimensional tight-binding lattice.
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Shuhei Nakamura et al. / Physics Procedia 27 (2012) 308 – 311
Fig. 2. The temperature T dependence of Josephson critical current IC for the junction with (a) odd LF (LF = 3) and (b) even LF (LF = 4). g is the band gap of FI. TC is the superconducting transition temperature. We set Δ = 0.01t.
in the matrix form [13]. Here h ( j, j ) = −t(δ j, j +1 + δ j, j −1 ), V = 2t + g is the exchange splitting of FI with g being the gap between spin up & down bands. The wave function of a bound state in FI is given by
fR f 0 βj 0 −β j ΨFI ( j) = (−1) j e−β j + L (−1) j eβ j + e + e , (6) gR gL 0 0 where β = cosh−1 (1 + E/2t + g/4t) is the imaginary part of the quasiparticle wave-number. By applying boundary conditions, ΨL (0) = ΨFI (0), ΨL (1) = ΨFI (1), ΨR (LF ) = ΨFI (LF ) and ΨR (LF + 1) = ΨFI (LF + 1), we can obtain the Andreev reflection coefficients rhe and reh analytically. The Josephson current can be calculated from the Furusaki-Tsukada formula [16, 17], IJ =
Δ e rhe − reh , kB T Ωn Ω
(7)
n
where T is the temperature and Ωn = ω2n + Δ2 with ωn = (2n + 1)πkB T being the Matsubara frequency. As long as we consider the case with g > Δ, the Josephson current phase relation is given by I J = IC sin φ. Thus we can compute IC from the relation IC = I J (φ = π/2). 3. Results In this section, we show the numerical results for the Josephson current I J in the finite temperature regime. Fig. 2 shows the temperature T dependence of the Josephson critical current IC for the cases that LF is (a) odd (LF = 3) and (b) even (LF = 4). In the zero-temperature (T TC ) limit, it is clear from Fig. 2 that IC is negative (positive) for odd (even) L. This observation is consistent with previous theoretical results, i. e., the atomic scale 0-π transition [13]. Importantly we have found the π-0 (0-π) transition by increasing the temperature T for the junction with odd (even) LF . This thermally induced 0-π transition becomes more prominent in the small-barrier case (g t). Therefore we can experimentally confirm the existence of the π state in S/FI/S junctions by simply observing the nonmonotonic T dependence of |IC |. The physical mechanism behind this anomalous phenomena will be given elsewhere. On the other hand, in the high-barrier limit (g t), we found that IC shows simple Ambegaokar-Baratoff type monotonic behavior and does not change its sign. This result indicates that the π (0) state is quite stable against the temperature variation. In an actual FI material, e.g., La2 BaCuO5 [18, 19], g/t ≈ 11 1 [13]. Thus we can expect a robust π state formation in junctions with these FI barriers. Such a robust property is very important for future electronic device applications.
Shuhei Nakamura et al. / Physics Procedia 27 (2012) 308 – 311
4. Conclusion To summarize, we have theoretically investigated the finite Temperature Josephson transport in a S/FI/S junction by solving the B-dG equation. The 0-π transition is induced by increasing the temperature T in the low barrier cases (g < t). This indicates that the existence of the π state can be experimentally confirmed by simply measuring the nonmonotonic T dependence of IC . In the high-barrier limit (g t), on the other hand, we found that the atomic-scale 0-π transition is robust against the T variation. Therefore, such FIbased Josephson junctions will be a promising candidate for a coherent and robust superconducting quantum bit as well as a classical superconducting digital circuit. 5. Acknowledgements We would like to thank Y. Asano for useful discussion. This work was supported by JST-CREST, the ”Topological Quantum Phenomena” (No. 22103002) KAKENHI on Innovative Areas, and a Grant-in-Aid for Scientific Research (No. 22710096) from MEXT of Japan. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A. I. Buzdin, Rev. Mod. Phys. 77 (2005) 935. A. A. Golubov, M. Y. Kupriyanov, E. Il’ichev, Rev. Mod. Phys. 76 (2004) 411. T. Kontos, M. Aprili, J. Lesueur, F. Genˆet, B. Stephanidis, R. Boursier, Phys. Rev. Lett. 89 (2002) 137007. V. V. Ryazanov, V. A. Oboznov, A. Y. Rusanov, A. V. Veretennikov, A. A. Golubov, J. Aarts, Phys. Rev. Lett. 86 (2001) 2427. L. B. Ioffe, V. B. Geshkenbein, M. V. Feigel’man, A. L. Fauch´ere, G. Blatter, Nature 398 (1999) 679. G. Blatter, V. B. Geshkenbein, L. B. Ioffe, Phys. Rev. B 63 (2001) 174511. S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, Physica C 437-438 (2006) 136. S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, A. A. Golubov, Phys. Rev. B 74 (2006) 180502(R). T. Ortlepp, Ariando, O. Mielke, C. J. M. Verwijs, K. F. K. Foo, H. Rogalla, F. H. Uhlmann, H. Hilgenkamp, Science 312 (2006) 1495. S. Kawabata, A. A. Golubov, Physica E 40 (2007) 386. A. K. Feofanov, V. A. Oboznov, V. V. Bolginov, J. Lisenfeld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko, M. Khabipov, D. Balashov, A. B. Zorin, P. N. Dmitriev, V. P. Koshelets, A. V. Ustinov, Nature Phys. 6 (2010) 593. S. Kawabata, Y. Asano, Int. J. Mod. Phys. B 23 (2009) 4329. S. Kawabata, Y. Asano, Y. Tanaka, A. A. Golubov, S. Kashiwaya, Phys. Rev. Lett. 104 (2010) 117002. S. Kawabata, Y. Asano, Low. Temp. Phys. 36 (2010) 915. K. Senapati, M. G. Blamire, Z. H. Barber, Nature Mat. 10 (2011) 849. A. Furusaki, M. Tsukada, Solid State Commun. 78 (1991) 299. A. Furusaki, M. Tsukada, Phys. Rev. B 43 (1991) 10164. F. Mizuno, H. Masuda, I. Hirabayashi, S. Tanaka, M. Hasegawa, U. Mizutani, Nature 345 (1990) 788. V. Eyert, K. H. H¨oc, and P. S. Riseborough, Europhys. Lett. 31 (1995) 385.
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