Defect Pinning and Critical Current of Magnetic Vortex Cluster in Mesoscopic Type-1.5 Superconductors

Abstract

Based on two-band time-dependent Ginzburg–Landau theory, we study the electromagnetic properties of mesoscopic type-1.5 superconductors with different defect configurations. We perform numerical simulations with the finite element method, and give direct evidence for the existence of a vortex cluster phase in the presence of nonmagnetic impurity. In addition, we also investigate the depinning critical current of the magnetic vortex cluster induced by the isotropic or anisotropic defect structure under the external current. Our theoretical results thus indicate that the diversity of impurity deposition has a significant influence on the semi-Meissner state in type-1.5 superconductors.

1. Introduction

Over the past two decades, two-band superconductivity has become an important research subject in condensed matter physics. This field started from the discovery of superconductivity in ${\mathrm{MgB}}_2$ [1], where the existence of two distinct superconducting gaps revealed the complexity of Fermi surface topology in the relevant system. Since then, extensive theoretical and experimental studies have been performed to provide novel insights into unconventional superconducting pairing mechanisms and physical properties in these materials. For example, multi-gap superconductivity signals a new pathway to achieve more superconducting pairing modes, which can induce phase competition or coexistence between multiple bands by adjusting the external magnetic field or impurity distribution. Furthermore, magnetic vortex behavior can be optimized through rational design of multi-band structures and its interaction with impurities can improve the overall performance of superconducting devices [2,3].

As we know, each condensate in two-band superconductors is predicted to support vortex excitation with fractional quantum flux [4,5]. Due to interband Josephson coupling, the vortices from different condensates are bound together with the string interaction and their normal cores will be locked to form a composite vortex with the standard integer quantum flux. Therefore, the vortex physics in two-band systems is influenced by the coherence lengths ${\xi }_1$ and ${\xi }_2$ as well as the magnetic field penetration depth $\lambda$. When the particular condition ${\xi }_1<\sqrt{2}\lambda <{\xi }_2$ is satisfied, a new superconducting state may be exhibited that combines characteristics of both type-1 and type-2 superconductors. This so-called semi-Meissner phase or vortex cluster phase is formed due to the interaction of long-range attraction and short-range repulsion between composite vortex excitations [6,7]. The existence of this novel vortex pattern was first visualized by Bitter decorations on high-quality ${\mathrm{MgB}}_2$ single crystals in 2009 [8]. Thereafter, zero-field muon spin experiments have also revealed the presence of this type-1.5 superconducting state in unconventional superconductors ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ [9,10] and ${\mathrm{LaPt}}_3\mathrm{Si}$ [11,12].

Recent seminal works of the Berger–Sardella group and Maksimova–Kashurnikov group on vortex dynamics and the critical current of mesoscopic superconductors have provided important inspiration for our theoretical research [13,14,15,16,17,18,19]. In the present paper, we study the electromagnetic effect of mesoscopic type-1.5 superconductors with different impurity distributions based on time-dependent Ginzburg–Landau (TDGL) theory. With the COMSOL Multiphysics software and the finite element method, our results directly show the crossover of this mesoscopic system from the diamagnetic Meissner state to the vortex cluster phase, and ultimately to the Abrikosov lattice phase. We also investigate the effects of isotropic and anisotropic defect structures on the pattern of magnetic vortex distribution. Furthermore, we discuss the depinning critical current of the magnetic vortex cluster induced by different pinning configurations under the external current. All of our theoretical results indicate that the diversity of impurity deposition has a significant influence on the collective behaviors of magnetic vortices in type-1.5 superconducting systems.

The rest of this article is organized as follows. In Section 2, we introduce the two-band TDGL theory and apply this formalism to type-1.5 superconductors. In Section 3, we describe the procedure of numerical simulations based on the finite element method. Then, in Section 4, we discuss the impurity effect and critical current of vortex clusters in the relevant mesoscopic systems. Finally, Section 5 offers a conclusion of the paper.

2. Theoretical Formalism

The simplest GL free energy functional of two-gap superconductors can be written as follows [20,21,22,23,24]:

$$\begin{array}{cc}\hfill F=& \sum _i\left[{\displaystyle \frac{1}{2m_i}}{\left|\mathsf{\Pi }{\mathsf{\Psi }}_i\right|}^2-{\alpha }_i{\left|{\mathsf{\Psi }}_i\right|}^2+{\displaystyle \frac{{\beta }_i}{2}}{\left|{\mathsf{\Psi }}_i\right|}^4\right]+{\gamma }_1{\left|{\mathsf{\Psi }}_1\right|}^2{\left|{\mathsf{\Psi }}_2\right|}^2\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_2}{2}}\left[{\left({\mathsf{\Psi }}_1^{*}{\mathsf{\Psi }}_2\right)}^2+\mathrm{c}.\mathrm{c}.\right]+{\displaystyle \frac{1}{2{\gamma }_3}}\left[{\left(\mathsf{\Pi }{\mathsf{\Psi }}_1\right)}^{*}·\mathsf{\Pi }{\mathsf{\Psi }}_2+\mathrm{c}.\mathrm{c}.\right]+{\displaystyle \frac{{\mathbf{B}}^2}{8\pi }}.\hfill \end{array}$$

(1)

Here, ${\mathsf{\Psi }}_i$$(i=1,2)$ represents the superconducting order parameter and $m_i$ is the effective mass for each band. ${\gamma }_1$ and ${\gamma }_2$ represent the interband quartic interactions, and ${\gamma }_3$ describes the gradient interaction between these two bands. The coefficient ${\alpha }_i$ is a function of temperature, while ${\beta }_i$ is independent of temperature. In the presence of impurity, the parameters ${\alpha }_1$ and ${\alpha }_2$ can be approximately expressed as ${\alpha }_i={\alpha }_{i0}f\left(\mathbf{r}\right)$. Here, we introduce a function $f\left(\mathbf{r}\right)$ between $+1$ and $-1$ to model the defect sites, which will deplete the superconducting state at specific positions [25,26]. We also define the covariant derivative operator $\mathsf{\Pi }=\left(-\mathrm{i}\hslash \nabla -2e\mathbf{A}/c\right)$ with the vector potential $\mathbf{A}$ and the magnetic field $\mathbf{B}=\nabla \times \mathbf{A}$.

If the superconductor is driven out of equilibrium, the order parameter should relax back to its equilibrium value. It is well known that this deviation of superconducting materials can be conveniently described by TDGL theories. The single-band TDGL equations were first proposed by Schmid [27] and are derived from the microscopic BCS theory by Gor’kov and Éliashberg [28]. The extension of these TDGL equations to multi-component superconducting systems can be written as follows [29,30]:

$$-{\mathsf{\Gamma }}_i{\displaystyle \frac{\partial {\mathsf{\Psi }}_i}{\partial t}}={\displaystyle \frac{\delta F}{\delta {\mathsf{\Psi }}_i^{*}}}\mathrm{and}-{\sigma }_n{\displaystyle \frac{\partial \mathbf{A}}{\partial t}}={\displaystyle \frac{\delta F}{\delta \mathbf{A}}}$$

(2)

where ${\mathsf{\Gamma }}_i$ is the relaxation time of order parameters and ${\sigma }_n$ represents the electrical conductivity of the normal sample in the two-band case. Therefore, minimization of the free energy F with respect to ${\mathsf{\Psi }}_i$ and $\mathbf{A}$ leads to the following dimensionless TDGL equations in the zero-electrostatic potential gauge:

$$-{\mathsf{\Gamma }}_1{\displaystyle \frac{\partial {\mathsf{\Psi }}_1}{\partial t}}={\mathsf{\Pi }}^2{\mathsf{\Psi }}_1+{\displaystyle \frac{m_1}{{\gamma }_3}}{\mathsf{\Pi }}^2{\mathsf{\Psi }}_2-\left[f\left(\mathbf{r}\right)-{\left|{\mathsf{\Psi }}_1\right|}^2-{\displaystyle \frac{{\gamma }_1}{{\beta }_1}}{\left|{\mathsf{\Psi }}_2\right|}^2\right]{\mathsf{\Psi }}_1+{\displaystyle \frac{{\gamma }_2}{{\beta }_1}}{\mathsf{\Psi }}_2^2{\mathsf{\Psi }}_1^{*},$$

(3)

$$\begin{array}{c}\hfill -{\mathsf{\Gamma }}_2{\displaystyle \frac{\partial {\mathsf{\Psi }}_2}{\partial t}}={\displaystyle \frac{m_1}{m_2}}{\mathsf{\Pi }}^2{\mathsf{\Psi }}_2+{\displaystyle \frac{m_1}{{\gamma }_3}}{\mathsf{\Pi }}^2{\mathsf{\Psi }}_1-\left[{\displaystyle \frac{{\alpha }_{20}}{{\alpha }_{10}}}f\left(\mathbf{r}\right)-{\displaystyle \frac{{\beta }_2}{{\beta }_1}}{\left|{\mathsf{\Psi }}_2\right|}^2-{\displaystyle \frac{{\gamma }_1}{{\beta }_1}}{\left|{\mathsf{\Psi }}_1\right|}^2\right]{\mathsf{\Psi }}_2+{\displaystyle \frac{{\gamma }_2}{{\beta }_1}}{\mathsf{\Psi }}_1^2{\mathsf{\Psi }}_2^{*}\end{array}$$

(4)

and

$$-{\displaystyle \frac{\partial \mathbf{A}}{\partial t}}={\kappa }_1^2\nabla \times \nabla \times \mathbf{A}-{\mathbf{j}}_s.$$

(5)

with the supercurrent

$$\begin{array}{c}\hfill {\mathbf{j}}_s=\mathrm{Re}\left({\mathsf{\Psi }}_1^{*}\mathsf{\Pi }{\mathsf{\Psi }}_1\right)+{\displaystyle \frac{m_1}{m_2}}\mathrm{Re}\left({\mathsf{\Psi }}_2^{*}\mathsf{\Pi }{\mathsf{\Psi }}_2\right)+{\displaystyle \frac{m_1}{{\gamma }_3}}\mathrm{Re}({\mathsf{\Psi }}_1^{*}\mathsf{\Pi }{\mathsf{\Psi }}_2+{\mathsf{\Psi }}_2^{*}\mathsf{\Pi }{\mathsf{\Psi }}_1).\end{array}$$

(6)

Here, in the clean limit with the nonmagnetic impurity function $f=1$, we at first introduce the coherence length ${\xi }_i^2={\hslash }^2/\left(2m_i{\alpha }_{i0}\right)$, the London penetration depth ${\lambda }^{-2}={\lambda }_1^{-2}+{\lambda }_2^{-2}$ with ${\lambda }_i^{-2}=4\pi e^2{\mathsf{\Psi }}_{i0}^2/\left(m_i{c}^2\right)$ and ${\mathsf{\Psi }}_{i0}=\sqrt{{\alpha }_{i0}/{\beta }_i}$, and the GL parameter ${\kappa }_1={\lambda }_1/{\xi }_1$. We then take the coordinate $\mathbf{r}$ in units of ${\xi }_1$, the time t in units of $t_0=m_1{\sigma }_n/\left(4e^2{\mathsf{\Psi }}_{10}^2\right)$, ${\mathsf{\Gamma }}_i$ in units of ${\alpha }_{10}{t}_0$, and the order parameter ${\mathsf{\Psi }}_i$ in units of ${\mathsf{\Psi }}_{10}$. We also set the magnetic field $\mathbf{B}$ in units of $H_0={\mathsf{\Phi }}_0/\left(2\pi {\xi }_1^2\right)$ with the flux quantum ${\mathsf{\Phi }}_0=\pi \hslash c/e$ and the vector potential $\mathbf{A}$ in units of $A_0=H_0{\xi }_1$.

Following ref. [6], multi-component systems allow a type of superconductivity that is distinct from type-1 or type-2 superconductors. With the condition ${\xi }_1<\sqrt{2}\lambda <{\xi }_2$, the type-1.5 superconducting state will originate from a peculiar vortex interaction that exhibits short-range repulsion and long-range attraction characteristics. The short-range repulsion prevents adjacent vortices from overlapping, while the long-range attraction facilitates the clustering of composite vortices. Consequently, this state is different from type-1 superconductors, which completely repel magnetic flux, and type-2 superconductors, which allow considerable magnetic flux penetration and the formation of a vortex lattice. In the ideal sample, the constraint mentioned above can be specifically expressed as

$$\sqrt{{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{m_1}{m_2}}{\displaystyle \frac{{\alpha }_{20}}{{\alpha }_{10}}}{\displaystyle \frac{{\beta }_1}{{\beta }_2}}\right)}<{\kappa }_1<\sqrt{{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{m_1}{m_2}}{\displaystyle \frac{{\alpha }_{10}}{{\alpha }_{20}}}+{\left({\displaystyle \frac{m_1}{m_2}}\right)}^2{\displaystyle \frac{{\beta }_1}{{\beta }_2}}\right]}.$$

(7)

In this circumstance, the magnetic composite vortices will form vortex clusters and coexist with domains of the two-component Meissner state in the framework of the GL theory. Furthermore, note that for the energy to be positively defined, the kinetic terms should give the relation $m_1{m}_2-{\gamma }_3^2<0$. Also, for the free energy functional to be bounded from below, the fourth-order terms in the condensates satisfy the constraint ${\beta }_1{\beta }_2-{({\gamma }_1+{\gamma }_2)}^2>0$ [31].

In order to numerically solve Equations (3)–(5), we need to specify appropriate boundary conditions of the superconducting sample. We use the following superconductor–insulator (or vacuum) boundary conditions [32,33,34]:

$$\nabla {\mathsf{\Psi }}_i·\mathbf{n}=0,\mathbf{A}·\mathbf{n}=0\mathrm{and}\nabla \times \mathbf{A}={\mathbf{H}}_e$$

(8)

where $\mathbf{n}$ is the outward unit vector normal to the boundary and the external applied magnetic field is set as ${\mathbf{H}}_e=H_e\widehat{\mathbf{z}}$. The first two conditions simply indicate that any current passing through the interface between a superconducting domain and vacuum/insulator would be nonphysical. The third equation represents the continuity of magnetic field across the boundary. The partial differential Equations (3)–(5) will be solved numerically for the mesoscopic geometry in the two-dimensional space. The initial conditions at $t=0$ are taken as $|{\mathsf{\Psi }}_i|=1$ and $\mathbf{A}=(0,0)$ on the $xy$-plane, corresponding to the Meissner state and zero magnetic field inside the superconductor.

In this investigation, we also study the vortex dynamics and critical current in the presence of impurity by applying an external current density ${\mathbf{j}}_n$ flowing through the sample in the $\widehat{\mathit{x}}$-direction. Following ref. [35], the current will be modeled by introducing an additional magnetic field $\mathsf{\Delta }\mathbf{B}$ in the numerical calculations. In the chosen $15{\xi }_1\times 15{\xi }_1$ geometry, this extra field is directed along the $\widehat{\mathit{z}}$-axis and is constructed so that the total magnitude of the applied field is $H_e-\mathsf{\Delta }H$ at $y=0$ and $H_e+\mathsf{\Delta }H$ at $y=15{\xi }_1$. From Equation (5), we have ${\mathbf{j}}_n=-\partial \mathbf{A}/\partial t={\kappa }_1^2\nabla \times \left(\mathsf{\Delta }\mathbf{B}\right)$, which can be explicitly written as ${\mathbf{j}}_n=j_n\widehat{\mathit{x}}$ with $j_n=2{\kappa }_1^2\mathsf{\Delta }H/\left(15{\xi }_1\right)$. Meanwhile, the electrical field is given by $\mathbf{E}=-(1/c)\partial \mathbf{A}/\partial t$ in the Weyl gauge. Then, the voltage V in the $\widehat{\mathit{x}}$-direction is $V=-{\int }_0^{15{\xi }_1}{E}_{x}dx=(1/c){\int }_0^{15{\xi }_1}(\partial A_x/\partial t)dx$ for the system. In the simulations, we set the normal current $j_n$ in units of $j_0=H_0/{\xi }_1$ and the sample voltage V in units of $V_0=H_0{\xi }_1^2/\left(c{t}_0\right)$.

At this point, we would like to briefly discuss the main limitation of the TDGL equations in multi-band superconducting systems. This phenomenological theory assumes that thermal excitations in the superconductor are slow-moving and in equilibrium with the local values of energy gaps and external fields. This occurs only in superconductors in which the characteristic interaction time between phonons and thermal excitations is faster than the characteristic time for normal to superfluid conversion [36].

3. Finite Element Method and Numerical Computations

Based on the COMSOL Multiphysics software platform [37], we will describe the procedure of the numerical simulations on the TDGL equations in this section. We first split the order parameters into the real and imaginary parts, i.e., ${\mathsf{\Psi }}_1=u_1+\mathrm{i}{u}_2$ and ${\mathsf{\Psi }}_2=u_3+\mathrm{i}{u}_4$. The magnetic potential is also written in component form as $\mathbf{A}=(u_5,u_6)$. In order to implement the boundary conditions, we will introduce an auxiliary variable $u_7(x,y,t)$ for reasons explained below. In the procedure of simulations, we set ${\mathsf{\Gamma }}_1={\mathsf{\Gamma }}_2=5$, ${\gamma }_1=0$ and $m_1=2m_2={\gamma }_3$. To stabilize the semi-Meissner state, we also take ${\alpha }_{10}={\alpha }_{20}$ and ${\beta }_1={\beta }_2=2{\gamma }_2$ in the calculations.

In this way, we can transform the TDGL equations into the general form of partial differential equations in this software package as follows:

$$\sum _k{\mu }_{jk}{\displaystyle \frac{\partial u_k}{\partial t}}+\sum _l{\partial }_l{\nu }_{jl}={\eta }_j.$$

(9)

Here, we have $j,k=1,2,\cdots ,7$, $l=1,2$ and $({\partial }_1,{\partial }_2)=({\partial }_x,{\partial }_y)$. The $7\times 7$ matrix ${\mu }_{jk}$ and the $7\times 2$ column vector ${\nu }_{jl}$ take the form

$${\mu }_{jk}=\left[\begin{array}{ccccccc}5& 0& 0& 0& 0& 0& 0\\ 0& 5& 0& 0& 0& 0& 0\\ 0& 0& 5& 0& 0& 0& 0\\ 0& 0& 0& 5& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(10)

and

$${\nu }_{jl}=\left[\begin{array}{cc}-u_{1x}-u_{3x}& -u_{1y}-u_{3y}\\ -u_{2x}-u_{4x}& -u_{2y}-u_{4y}\\ -u_{1x}-2u_{3x}& -u_{1y}-2u_{3y}\\ -u_{2x}-2u_{4x}& -u_{2y}-2u_{4y}\\ 0& {\kappa }_1^2\left(u_{6x}-u_{5y}-H_e\right)\\ {\kappa }_1^2\left(u_{5y}-u_{6x}+H_e\right)& 0\\ u_5& u_6\end{array}\right].$$

(11)

Here, we note that the subscript x or y denotes the partial derivative with respect to the corresponding variable. Meanwhile, the driving force ${\eta }_j$ contains all other terms in the TDGL equations except the left-hand side of Equation (9), and detailed calculations will give all the components explicitly as

$$\begin{array}{cc}\hfill {\eta }_1& =f\left(\mathbf{r}\right)u_1-{\displaystyle \frac{1}{2}}\left(2u_1^2+2u_2^2+u_3^2-u_4^2+2u_5^2+2u_6^2\right)u_1+\left(u_{5x}+u_{6y}\right)u_2-\left(u_5^2+u_6^2\right)u_3\hfill \\ \hfill & +\left(u_{5x}+u_{6y}\right)u_4+2\left(u_{2x}+u_{4x}\right)u_5+2\left(u_{2y}+u_{4y}\right)u_6-u_2{u}_3{u}_4,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\eta }_2& =f\left(\mathbf{r}\right)u_2-{\displaystyle \frac{1}{2}}\left(2u_1^2+2u_2^2-u_3^2+u_4^2+2u_5^2+2u_6^2\right)u_2-\left(u_{5x}+u_{6y}\right)u_1-\left(u_{5x}+u_{6y}\right)u_3\hfill \\ \hfill & -\left(u_5^2+u_6^2\right)u_4-2\left(u_{1x}+u_{3x}\right)u_5-2\left(u_{1y}+u_{3y}\right)u_6-u_1{u}_3{u}_4,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\eta }_3& =f\left(\mathbf{r}\right)u_3-{\displaystyle \frac{1}{2}}\left(u_1^2-u_2^2+2u_3^2+2u_4^2+4u_5^2+4u_6^2\right)u_3-\left(u_5^2+u_6^2\right)u_1+\left(u_{5x}+u_{6y}\right)u_2\hfill \\ \hfill & +2\left(u_{5x}+u_{6y}\right)u_4+2\left(u_{2x}+2u_{4x}\right)u_5+2\left(u_{2y}+2u_{4y}\right)u_6-u_1{u}_2{u}_4,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\eta }_4& =f\left(\mathbf{r}\right)u_4-{\displaystyle \frac{1}{2}}\left(u_2^2-u_1^2+2u_3^2+2u_4^2+4u_5^2+4u_6^2\right)u_4-\left(u_{5x}+u_{6y}\right)u_1-\left(u_5^2+u_6^2\right)u_2\hfill \\ \hfill & -2\left(u_{5x}+u_{6y}\right)u_3-2\left(u_{1x}+2u_{3x}\right)u_5-2\left(u_{1y}+2u_{3y}\right)u_6-u_1{u}_2{u}_3,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\eta }_5& =\left(u_{2x}+u_{4x}\right)u_1-\left(u_{1x}+u_{3x}\right)u_2+\left(u_{2x}+2u_{4x}\right)u_3-\left(u_{1x}+2u_{3x}\right)u_4\hfill \\ \hfill & -\left(u_1^2+u_2^2+2u_3^2+2u_4^2+2u_1{u}_3+2u_2{u}_4\right)u_5,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\eta }_6& =\left(u_{2y}+u_{4y}\right)u_1-\left(u_{1y}+u_{3y}\right)u_2+\left(u_{2y}+2u_{4y}\right)u_3-\left(u_{1y}+2u_{3y}\right)u_4\hfill \\ \hfill & -\left(u_1^2+u_2^2+2u_3^2+2u_4^2+2u_1{u}_3+2u_2{u}_4\right)u_6,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\eta }_7& =u_{5x}+u_{6y}+u_7.\hfill \end{array}$$

(18)

Now, we can examine the boundary conditions under this formalism. With the normal vector $\mathbf{n}=(n_1,n_2)$ and the column vector ${\nu }_{jl}$, the boundary conditions in Equation (8) can be simply cast into the compact form

$$\sum _l{n}_l{\nu }_{jl}=0$$

(19)

which is best suited to the COMSOL Multiphysics simulations. We also note that from the last equation ($j=7$) in (9), our manipulations will give a trivial solution $u_7=0$ for this auxiliary variable, ensuring the self-consistency of our problem.

COMSOL Multiphysics is a versatile and advanced simulation platform that is designed to tackle complex engineering and scientific problems. Its core principle is to numerically solve partial differential equations based on the finite element method [38,39,40]. The process begins with discretizing the computational domain and subdividing the lattice cell into small subregions called elements. Triangular elements are preferred due to their flexibility in handling complex and irregular shapes. The process will transform the continuous domain into a finite element mesh and enable precise numerical computations. Following this step, a function space typically composed of piecewise continuous polynomials is constructed to ensure smoothness across element boundaries. Subsequently, Lagrangian shape functions are selected as basis functions for their ability to achieve high computational accuracy and numerical stability [41]. Finally, the software employs an implicit solver that typically incorporates consistent initialization of the backward Euler method to ensure significant stability for time-dependent simulations. In our numerical computations, we take the time step $\mathsf{\Delta }t=0.5t_0$ and the relative tolerance ${10}^{-8}$ to control the convergence of the transient calculations for our system.

4. Results and Discussions

In this section, we will discuss the effect of impurity on the patterns of magnetic vortex distribution and the critical current for vortex cluster depinning in a 15$\xi \times 15\xi$ type-1.5 superconductor. Following Refs. [25,26], we have chosen the impurity function f to take the phenomenological form

$$f\left(\mathbf{r}\right)=\left\{\begin{array}{c}\begin{array}{cc}\hfill -0.5,& \mathrm{if}|\mathbf{r}-{\mathbf{r}}_0|<R\left|cos\left[p\left(\theta +{\displaystyle \frac{\pi }{4}}\right)\right]\right|\hfill \\ \hfill 1,& \mathrm{otherwise}\hfill \end{array}.\hfill \end{array}\right.$$

(20)

It is easy to see that the impurity is centered at ${\mathbf{r}}_0=(x_0,y_0)$, and its shape depends on the angle $\theta$ and different integer values of p. This means that the defect sites can be isotropic with radius R when $p=0$ and anisotropic with 2p-fold symmetry at $p\ne 0$. We take $R=0.5{\xi }_1$ for each pinning state in the simulations.

Meanwhile, based on Inequality (7), we expect to discover the semi-Meissner state within $1.22<{\kappa }_1<1.73$ in the clean limit. According to Ref. [42], the introduction of impurity into the material will enhance the effective magnetic penetration depth, thus leading to a larger effective GL parameter. Therefore, this allows the magnetic vortex phases to be observed at slightly smaller ${\kappa }_1$ values in the presence of impurity in type-1.5 superconductors. We set the value ${\kappa }_1=1.35$ for the following numerical calculations.

There are three main methods of trapping the magnetization in common use: zero field cooling (ZFC), field cooling (FC), and pulsed field magnetization (PFM). In our investigation, a specific numerical technique is applied to model the complete process of FC magnetization. This scheme is based on the two-dimensional $\mathbf{H}$-formulation and the commercial software package COMSOL Multiphysics 4.3a [37]. The ac/dc module of COMSOL is employed for the electromagnetic analysis and the heat transfer module is used for the thermal analysis, which are coupled together to carry out the calculations of magnetic flux density in the FC regime. In the FC process, the magnetic field is applied to the superconducting sample at $T>T_c$, which is then cooled below $T_c$. We thus assume a smooth transition from the normal state to the superconducting state to avoid non-convergence around the transition temperature.

To verify the availability of the method, we first take the impurity function with $N=1$ and $p=0$ and insert this pinning site at the center of the superconducting square. We then plot the magnetic flux density $B_z=u_{6x}-u_{5y}$ in units of $H_0$ (a–c) and the order parameter of the first condensate $\left|{\mathsf{\Psi }}_1\right|=\sqrt{u_1^2+u_2^2}$ in units of ${\mathsf{\Psi }}_{10}$ (d–f) at $t={10}^4{t}_0$ in Figure 1. With the external magnetic field $H_e$ taken as 0.25$H_0$, 0.55$H_0$, and 0.85$H_0$ sequentially, we can clearly observe the crossover of this type-1.5 system from the perfect diamagnetism state to the vortex cluster phase, and ultimately to the Abrikosov vortex lattice. Our numerical simulations also show that the cluster phase presents the vortex pattern with octagonal symmetry and appears in the region of $0.45H_0<H_e<0.72H_0$. Moreover, it is easy to see that the isotropic defect induces localized distortions of the flux lattice without breaking the $C_4$ rotational symmetry of the superconducting square.

Furthermore, we perform the simulations on this mesoscopic type-1.5 system with an anisotropic defect. For $N=1$ and $p=2$, we still take the impurity site at the center of the superconducting square and plot the $B_z$ and $\left|{\mathsf{\Psi }}_1\right|$ at $t={10}^4{t}_0$ in Figure 2. By setting $H_e$ as 0.55 $H_0$ and 0.85 $H_0$, we can observe the novel vortex cluster with $C_4$ (not $C_8$ in isotropic impurity case) symmetry shown in Figure 2b,e and the distorted flux lattice in Figure 2c,f, respectively. Our numerical data also indicate that the vortex cluster phase exists in the regime $0.35H_0<H_e<0.60H_0$ for this mesoscopic superconductor. As we see, the anisotropic defect occupies the smaller normal area compared with its isotropic counterpart, leading to a smaller $H_e$ for the emergence of the magnetic vortex phases.

At this stage, we also perform numerical computations on the depinning critical current of the magnetic vortex cluster in the presence of an isotropic or anisotropic defect structure. For $H_e=0.55H_0$ and ${\kappa }_1=1.35$, we plot the sample voltage V as a function of external current $j_n$ for different impurity configurations in Figure 3. From Figure 3, we can see that the system remains in the stable vortex cluster phase with zero voltage across the boundary until the threshold current is reached at $j_{c1}=0.3j_0$ (the isotropic impurity case) and $j_{c2}=0.15j_0$ (the anisotropic impurity case), respectively. With the further increase in current, the voltage increases from zero due to the energy dissipation of vortex motions in the depinning phase. In addition, the numerical result $j_{c1}>j_{c2}$ is in agreement with our theoretical expectation that the isotropic defect possesses a larger effective normal area and will exhibit a stronger pinning force on vortices than the anisotropic counterpart.

5. Conclusions

Based on two-band TDGL theory, we investigated the impurity effect on vortex collective behaviors in mesoscopic type-1.5 superconductors. With the finite element method, our numerical results show a clear signature for the existence of a vortex cluster phase in the presence of nonmagnetic impurity in the studied system. We also numerically calculated the depinning critical current of the magnetic vortex cluster induced by an isotropic or anisotropic defect structure under the external current. We hope that our theoretical results will inspire further research on better understanding novel vortex patterns and phase transitions in two-band superconductors.