Twin boundary effects on spontaneous half-quantized vortices in superconducting composite structures (d-dot’s)

Physica C xxx (2015) xxx–xxx

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Twin boundary effects on spontaneous half-quantized vortices in superconducting composite structures (d-dot’s) Norio Fujita a,⇑, Masaru Kato a, Takekazu Ishida b a b

Department of Mathematical Sciences, Osaka Prefecture University, 1-1, Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan Department of Physics and Electronics, Osaka Prefecture University, 1-1, Gakuencho, Nakaku, Sakai, Osaka 599-8531, Japan

a r t i c l e

i n f o

Article history: Received 3 February 2015 Received in revised form 3 April 2015 Accepted 8 April 2015 Available online xxxx Keywords: d-wave superconductivity Half-quantized vortex Ginzburg–Landau equations Finite element method Twin boundary Anisotropic effective mass

a b s t r a c t We investigate effects of anisotropy of an orthorhombic structure in twin domains on spontaneous half-quantized vortices (SHQVs) in a d-dot, which is a nano-scaled composite structure that consists of a d-wave superconductor embedded in an s-wave matrix. Since YBa2Cu3O7d (YBCO) has the orthorhombic structure, there are twin domains separated by twin boundaries (TBs). In order to analyze effects of TBs on SHQVs, we derive two-component Ginzburg–Landau equations, in which electrons in YBCO have anisotropic effective mass. It is found that the magnitude of field around SHQVs is monotonically decreasing with increasing anisotropy of effective mass and finally peak values of the fields become zero. This means that anisotropy of the effective mass suppresses SHQVs and when the anisotropy is too large, SHQVs do not appear in d-dot’s. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction A d-dot is a nano-scaled composite structure that consists of a d-wave superconductor embedded in an s-wave matrix [1,2]. Because phase difference of the d-wave superconducting order parameter between x- and y-directions in a momentum space is p due to its d-wave symmetry, corner junctions between d- and s-wave superconductors are called p-junctions. At each corner junction, compensating this phase difference, there spontaneously appears a half-quantized vortex without external magnetic fields [3]. Because of broken time-reversal symmetry, there are two stable states. Since we may make a superposition of these two states, lowering the size of the d-dot and increasing a transition probability between two states, such the d-dot may work as a quantum bit (qubit) in quantum computers [4]. There are many candidates for qubit, such as flux qubits [5–7], charge qubits [8–10], the Cooper pair box [11–13], but we believe that d-dot’s have considerable potential as qubits, because d-dot’s have advantages in scalability and stability. Fujii et al. made d-dot’s that consist of YBa2Cu3O7d (YBCO) and Pb, but they did not observe SHQVs [14]. One of reasons of this result may be defects in YBCO. YBCO has an orthorhombic structure, because CuO chains break tetragonal symmetry. CuO chains have two possible orientations and lead to formation of twin domains separated by twin ⇑ Corresponding author.

boundaries (TBs). Smilde et al. reported TBs suppress the spontaneous half-quantized vortices (SHQVs) [15]. To investigate how TBs affect on SHQVs, phenomenologically, we first derive Ginzburg–Landau (GL) equations for orthorhombic structured superconductor and solve them numerically using the finite element method (FEM) [1]. 2. Two-component Ginzburg–Landau equations with an anisotropic effective mass In order to incorporate effects of TBs into the GL equations, we introduce an anisotropic effective mass into the Gor’kov equations [16] ( )  2  2 1 @ e 1 @ e ~ x0 ; xn Þ ihxn  ih þ Ax  ih þ Ay þ l Gðx; 2mx @x c 2my @y c Z 00 ð1Þ þ dx Dðx; x00 ÞF y ðx00 ; x0 ; xn Þ ¼ dðx  x0 Þ; (

)  2  2 1 @ e 1 @ e ihxn  ih þ Ax  ih þ Ay þ l F y ðx; x0 ; xn Þ 2mx @x c 2my @y c Z 00  e 00 ; x0 ; xn Þ ¼ 0;  dx D ðx; x00 Þ Gðx ð2Þ

~ and F are the single particle normal and anomalous Green’s where G functions, respectively. mx and my are the effective mass along

http://dx.doi.org/10.1016/j.physc.2015.04.002 0921-4534/Ó 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: N. Fujita et al., Twin boundary effects on spontaneous half-quantized vortices in superconducting composite structures (d-dot’s), Physica C (2015), http://dx.doi.org/10.1016/j.physc.2015.04.002

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N. Fujita et al. / Physica C xxx (2015) xxx–xxx

we derive self-consistent equations for s- and d-wave order parameters, Ds and Dd ,

   c      Vs 2e hxD Vs eF 2 eF 2   2kd Ds ¼ 2kd Ds ln a Px þ Py Ds Vd pkB T Vd 2mx 2my    eF 2 eF 2  1  þ ð5Þ Px  Py Dd þ jDs j2 Ds þ jDd j2 Ds þ D2 d Ds ; 2 4mx 4my  c    2e hxD eF 2 eF 2   2kd a Dd ¼ 2kd Dd ln Px þ Py Dd pkB T 4mx 4my    eF 2 eF 2  3 1 þ Px  Py Ds þ jDs j2 Dd þ jDd j2 Dd þ D2 s Dd : ð6Þ 8 2 4mx 4my

Fig. 1. A calculation model of a d-dot.

x- and y-axis, respectively. xn ¼ ð2n þ 1ÞpT is the Matsubara frequency. The order parameter in real space is given

X D ðx; x0 Þ ¼ Vðx  x0 ÞT F y ðx; x0 ; xn Þ;

ð3Þ

xn

   ^2  k ^2 k ^02 ; ^02  k Vðx  x Þ ¼ V s þ V d k x y x y 0

Here c is the Euler constant, and xD is a cutoff frequency for interactions.

where Vðx  x0 Þ is an effective pairing interaction between two electrons, and V s and V d are interaction constants for s- and d-wave pairings [17], respectively. Substituting Eqs. (1)–(3) into Eq. (4),

kd ¼ Nð0ÞV d =2,

a ¼ 7fð3Þ=8ðpkB TÞ2 ,

eF ¼

2 2  2 k2Fx =2mx ¼  h h kFy =2my , where Nð0Þ is a density of states at the

Fermi

ð4Þ

Also surface,

2 2 ð h kx =2mx

P ¼ ihrR  ð2e=cÞAR ,

R ¼ ðx þ x0 Þ=2

and

2 2  h ky =2my Þ

nk ¼ þ  l. Note that gradient terms and anisotropic coupling terms between Ds and Dd in Eqs. (5) and (6) contain anisotropic mass. In order to understand how these terms affect SHQVs, we solve Eqs. (5) and (6) using the FEM [1].

Fig. 2. Distributions of the d-wave order parameter, three-dimensional surface plots and contour plots of magnetic field distributions for mx =my ¼ 1:0 ((a), (d) and (g)), mx =my ¼ 3:0 ((b), (e) and (h)) and mx =my ¼ 5:0 ((c), (f) and (i)), respectively. In (e) and (f), red and blue mean Hz > 0 and Hz < 0, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Please cite this article in press as: N. Fujita et al., Twin boundary effects on spontaneous half-quantized vortices in superconducting composite structures (d-dot’s), Physica C (2015), http://dx.doi.org/10.1016/j.physc.2015.04.002

N. Fujita et al. / Physica C xxx (2015) xxx–xxx

Fig. 3. Peak values of the magnetic field as functions of effective mass ratios for 0.5d0 (red circle), 0.75d0 (blue square), d0 (green rhombus), 1.5d0 (orange triangle) and 1.75d0 (purple triangle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Model of a d-dot In this calculation, we use material parameters of Pb and YBCO for s- and d-wave superconductors (SCs), respectively. Fig. 1 shows a calculation model of a d-dot. We consider following system; an s-wave matrix is L  LðL ¼ ns0 ¼ 50nd0 ’ 80 nmÞ and an embedded d-wave region is L=4  L=4ðL=4 ’ 20 nmÞ. Here, ns0 and nd0 are the coherence length of s- and d-wave SCs at 0 K. We consider a boundary region between s- and d-wave SCs and we set boundary thickness is d0 ¼ L=100 ’ 0:8 nm. Experimentally, a Au layer is sputtered between d- and s-wave regions to protect the d-wave SC from direct chemical reactions with the s-wave SC. In addition, there may be damaged regions on edges of the d-wave SC region during fabrication processes of the d-dot. The boundary region represents either the Au layer or the damaged region. We consider that the s-wave order parameter exists in the s-wave SC region and penetrates only into the boundary region but not into d-wave SC region. On the other hand, the d-wave order parameter exists in the d-wave SC region and penetrates only into the boundary region but not into the s-wave SC region. So, interactions between s- and d-wave order parameters, which are important for appearance of SHQVs, occur in the boundary region. 4. Numerical results and discussions In Fig. 2, the d-wave order parameter, and the three-dimensional surface plots and contour plots of magnetic field distributions are shown for isotropic system with mx =my ¼ 1:0 ((a), (d) and (g)), and anisotropic system with mx =my ¼ 3:0 ((b), (e) and (h)), mx =my ¼ 5:0 ((c), (f) and (i)), respectively. In Fig. 2(g), (h) and (i), red and blue colors mean Hz > 0 and Hz < 0, respectively. Comparing Fig. 2(a), (b) and (c), we see that shapes of d-wave order parameter distributions are almost same. Also comparing Fig. 2(d), (e) and (f), we see whole shapes of magnetic fields Hz are similar. Especially, SHQVs are in a same antiferromagnetic configuration, which is a typical order and experimentally showed by Hilgenkamp et al. [3]. But we notice that peak values of Hz in Fig. 2(e) and (f) are smaller than that of Fig. 2(d). Fig. 2(e) and (f) show that areas of SHQVs become large when mx =my is large. This means that magnitudes of SHQVs are suppressed by large mass anisotropy. For the SHQV, only a fluxxoid that consist of a contour integral of a supercurrent and a magnetic flux is quantized as a half of U0 ¼ hc=2e. When the anisotropy is large, contributions of the supercurrent is dominant, and therefore the magnetic field becomes small. To investigate relations between the magnitude of the magnetic field of SHQVs and the anisotropy, peak values of the magnetic

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field as a function of effective mass ratios are plotted in Fig. 3 for different boundary thicknesses. We can find the magnetic field of SHQVs is monotonically decreasing with increasing the anisotropy of effective mass. For the boundary thickness d = 0.8 nm case, the peak value of SHQVs becomes zero, when effective mass ratio exceeds 10. Therefore, when the anisotropy is too large, the SHQVs do not appear in d-dot’s. Fig. 3 also shows that peak values of SHQVs depend on the boundary thickness. If the boundary thickness d is thicker than d0, the peak values of SHQVs are small and suddenly decrease with increasing anisotropy of effective mass. For example, when the boundary thickness is 1.5d0, the peak value becomes to zero at mx =my ¼ 3:75. On the other hand, when the thickness is less than d0, the peak value do not rapidly decrease with increasing the effective mass ratio. For example, when the thickness is 0.75d0, at mx =my ¼ 10, the peak value is more than a half of that at mx =my ¼ 1. If the boundary becomes thinner, we may suppress effects of the anisotropy of effective mass and SHQVs become more stable. 5. Conclusions In order to analyze the effects of TBs on SHQVs, phenomenologically, we derive the two-component GL equations, in which electrons in YBCO have anisotropic effective mass. In this GL equations, gradient terms and anisotropic coupling terms between s-wave and d-wave superconducting order parameters become anisotropic. Solving these self-consistent equations numerically using the FEM, we find that effective mass anisotropy suppress SHQVs. And we also find that if the boundary is too thick, the SHQVs are not induced. In future, we will analyze how affect TBs on SHQVs using this model. Acknowledgements We would like to thank Y. Higashi and M. Umeda for beneficial discussions. This work is supported by the Program for Leading Graduate Schools of the Ministry of Education, Culture, Sports, Science and Technology in Japan (MEXT) and JPSP KAKENHI Grant Number 26400367. References [1] M. Kato, T. Ishida, T. Koyama, M. Machida, Superconductors – Materials, Properties and Applications, InTech, Croatia, 2012. pp. 319–342. [2] T. Ishida, M. Fujii, T. Abe, M. Yamamoto, S. Miki, S. Kawamata, K. Satoh, T. Yotsuya, M. Kato, M. Machida, T. Koyama, T. Terashima, S. Tsukui, M. Adachi, Physica C 437 (2006) 104–110. [3] H. Hilgenkamp, Ariando, H.-J.H. Smilde, D.H.A. Blank, G. Rijnders, H. Rogalla, J.R. Kirtley, C.C. Tsuei, Nature 422 (2003) 50–53. [4] T. Koyama, M. Machida, M. Kato, T. Ishida, Physica C 426–431 (2005) 1561– 1565. [5] T. Yamamoto, K. Inomata, K. Koshino, P.-M. Billangeon, Y. Nakamura, J.S. Tsai, New J. Phys. 16 (2014) 015017. [6] P.A. Volkov, M.V. Fistul, Phys. Rev. B 89 (2014) 054507. [7] R.H. Koch, G.A. Keefe, F.P. Milliken, J.R. Rozen, C.C. Tsuei, J.R. Kirtley, D.P. DiVincenzo, Phys. Rev. Lett. 96 (2006) 127001. [8] G. Cao, H-O. Li, T. Tu, L. Wang, C. Zhou, M. Xiao, G-C. Guo, H-W. Jiang, G-P. Guo, Nat. Commun. 4 (2013) 1401–1406. [9] G. Shinkai, T. Hayashi, T. Ota, T. Fujisawa, Phys. Rev. Lett. 103 (2009) 056802. [10] Y. Dovzhenko, J. Stehlik, K.D. Petersson, J.R. Petta, H. Lu, A.C. Gossard, Phys. Rev. B 84 (2011) 161302. [11] Y. Nakamura, Y.A. Pashkin, J.S. Tsai, Nature 398 (1999) 786–788. [12] Y.A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D.V. Averin, J.S. Tsai, Nature 421 (2003) 823–826. [13] T. Yamamoto, Y.A. Pashkin, O. Astafiev, Y. Nakamura, J.S. Tsai, Nature 425 (2003) 941–944. [14] M. Fujii, T. Abe, H. Yoshikawa, S. Miki, S. Kawamata, K. Satoh, T. Yotsuya, M. Kato, M. Machida, T. Koyama, T. Terashima, S. Tsukui, M. Adachi, T. Ishida, Physica C 426 (2005) 104–107. [15] H.J.H. Smilde, Ariando, D.H.A. Blank, G.J. Gerritsma, H. Hilgenkamp, H. Rogalla, Phys. Rev. Lett. 88 (2002) 057004. [16] L.P. Gor’kov, Sov. Phys. JETP 9 (1960) 1364–1367. [17] Y. Ren, J.H. Xu, C.S. Ting, Phys. Rev. Lett. 74 (1995) 3680–3683.

Please cite this article in press as: N. Fujita et al., Twin boundary effects on spontaneous half-quantized vortices in superconducting composite structures (d-dot’s), Physica C (2015), http://dx.doi.org/10.1016/j.physc.2015.04.002