Fractional vortices in a nano-scaled superconducting composite structure (d-dot) with a twin boundary

In order to investigate effects of twin boundary (TB) on structure of spontaneous half-quantized vortices (SHQVs) in a nano-scaled composite structure that is consisted of d- and s-wave superconductors (d-dot), we investigate magnetic field distribution with a d-dot model which has a single TB using anisotropic two-component Ginzburg-Landau equations and finite element method. We find that fractional vortices appear at the edges of TB and that the fractional vortices merge with SHQVs when effective interaction between d- and s-wave SCs is small.


Introduction
Corner junctions made with d-and s-wave superconductors (SCs) are called -junctions, because the phase between d-and s-wave SCs is different by due to symmetry of order parameter in d-wave SC [1]. With quantization of fluxoids including this phase differences, half-quantized vortices spontaneously appear around -junctions with zero magnetic field [2].
A d-dot, which is a nano-scaled composite structure that consists of a d-wave SC embedded in an s-wave matrix [3,4] has four -junctions and then four spontaneous half-quantized vortices (SHQVs). If the size of the d-wave SC in d-dot is sufficiently small than penetration depth of the s-wave SC, SHQVs form anti-ferromagnetic configuration. So d-dot has two arrangements of SHQVs, which are doubly degenerated stable states controlling by an external field or current [5,6] we can use it as a classical bit with. It is also predicted that we can use superposition of these stable states as quantum bit by reducing the size of d-dot's to increase transition probability between both states [7].
Fujii et al. made d-dot's that consist of YBa 2 Cu 3 O 7-δ (YBCO) and Pb, but SHQVs were not observed [8]. One of reasons of this result seems to be defects in YBCO, especially twin boundaries (TBs). Smilde et al. reported TBs suppress the SHQVs [9].
In this paper, to clarify effects of TB on structures of SHQVs in a d-dot, we investigate magnetic field distribution with a d-dot model that has a single TB using anisotropic two-component Ginzburg-Landau (GL) equations and finite element method (FEM). We show that fractional vortices appear at the edges of TB and that the fractional vortices merge with SHQVs when effective interaction between d-and swave SCs is small.

Methods
In order to incorporate effects of TBs into the GL equations, we introduce an anisotropic effective mass into the Gor'kov equations [10] as follows.
where and F are the single particle normal and anomalous Green's functions, respectively. m x and m y are the effective mass along x-and y-axis, respectively.
is the Matsubara frequency. The order parameter in real space is given where is an effective pairing interaction between two electrons in momentum space, and V s and V d are interaction constants for s-and d-wave pairings [11], respectively.
Substituting equations (1), (2) and (4) into equation (3), we derive self-consistent equations for sand d-wave order parameters [12], and Δ d , Here is the Euler constant, and is a cutoff frequency for interactions. Also , and where is a density of states at the Fermi surface. And where .
Note that gradient terms and anisotropic coupling terms between and in equations (5) and (6) contain anisotropic mass. In order to analyze how these terms affect d-dot's, we solve equations (5) and (6) using the FEM [3].
In this calculation, we use material parameters of YBCO and Pb for d-and s-wave SCs, respectively.    Figure 2 shows magnetic field distributions for isotropic system with m y /m x = 1.0 (a), and for anisotropic system with m y /m x = 2.0 (b), respectively. In figures 2 (a) and (b), red and blue colors mean H z < 0 and H z > 0, respectively. In this case we set ratios of V s and V d in d-wave SC (V s /V d ) d = 1.0 and in s-wave SC (V d /V s ) s = 1.0, respectively.

Results and Discussions
In the isotropic case, figure 2 (a), only four SHQVs appear on each corners of d-wave domain and form antiferromagnetic configuration [3]. In the anisotropic cases, figure 2 (b), shapes and configuration of magnetic field distributions are almost same as figure 2 (a), but we can also see additional vortices in solid circles (green circles). These additional vortices are fractional vortices, not SHQVs, because peak values of these additional vortices are less than SHQVs' in figure 2 (a). One can understand how these fractional vortices appear as below. The distributions of supercurrents around the corners are in an elliptic form, because the penetration depth is inversely proportional to the square root of effective mass Here e * and m * is charge and effective mass of cooper pair i.e. e * =2e, m * =m x or m y . Around the corner junctions, supercurrent j s (r) is large. Then they decrease continuously from the corner to the middle point of the corners. Because penetration depth changes due to the change of m * across the TB, there is finite imbalance of j s (r) at the edges of the TB. This supercurrent imbalance along the boundary of dand s-wave SCs, especially in solid circles (green circles), generate fractional vortices.
To analyze dependence of interaction between d-and s-wave SCs, we obtain magnetic field distribution for the case of (V s /V d ) d = (V d /V s ) s = 0.1, respectively.

Summary
In conclusion, in order to analyze the effects of TBs on structure of magnetic field distribution in d-dot, phenomenologically, we calculated magnetic field distribution in a d-dot model with a single TB using modified two-component GL equations that include anisotropic mass terms and the FEM. We find that fractional vortices appear at the edges of TB beside SHQVs and that the fractional vortices merge with the nearest parallel SHQVs when effective interaction between d-and s-wave SCs is small.