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Superconducting 3D-sample with general boundary conditions in a tilted magnetic field
Physica C: Superconductivity and its applications 558 (2019) 1–6
Contents lists available at ScienceDirect
Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc
Superconducting 3D-sample with general boundary conditions in a tilted magnetic field
T
J. Barba-Ortega ,a, M.R. Joyaa, E. Sardellab ⁎
a b
Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia Departamento de Física, Universidade Estadual Paulista, UNESP, Baurú, SP, Brazil
ARTICLE INFO
ABSTRACT
Keywords: Vortex Ginzburg–Landau Magnetization Mesoscopics Superconductor Tilded field
We consider a mesoscopic superconducting three-dimensional cube immersed in a tilted magnetic field H oriented along an angle θ with respect to the z-axis, in the xz plane. The sample is in contact with a metallic material or with another superconductor in the xz and yz planes, while the z = 0 plane remains in contact with a dielectric material. These boundary conditions are simulated by the de Gennes extrapolation length b. We analyzed the magnetic induction, the superconducting electron density, and the magnetization curves as functions of H for different values of b on the lateral surfaces of the sample. We show that the magnetization and the vortex configuration depend on θ and b. An analytical linear and exponential dependence of the maximum of the magnetization on θ and b, respectively, was found.
PACS: 74.20.De 74.25.Bt 74.25.Uv 74.25.Wx
1. Introduction As is well known, in spatially homogeneous anisotropic superconducting samples, the Shubnikov vortex lattice consists of rectilinear magnetic quantum flux aligned along the external applied magnetic field. The magnetic energy depends on the orientation of the magnetic field with respect to the crystallographic axis [1–3]. Several theoretical and experimental studies of superconductors systems in the presence of magnetic fields have been performed in the last few decades. For example, Feinberg et al. assumed that the vortices in this magnetic field have a kink structure and the first critical magnetic field depends on the angle [4]. Buzdin et al. studied the angular and field dependences of the magnetization of superconductors in tilted magnetic fields. They discussed the inclined Abrikosov vortex lattice in quasi one- and two-dimensional superconductors and found that in contrast to the typical situation with one maximum of the magnetization at the first vortex entry magnetic field, an additional maximum appears in the region of the higher field [5]. Grigorieva et al., observed the vortex structure experimentally using the Bitter technique in YBaCuO and YBa2 (Cu1 x Alx )3 O7 single crystals. They found that the vortex configuration remained unchanged for an interval of tilt angles θ > 25 with respect to the c-axis [6,7]. Ivlev et al. showed that at high Tc, superconductors with a well-pronounced structure of the order parameter, a
⁎
magnetic field tilted to the superconducting planes gives rise to a system of kinks on the vortex lines. This leads to strong anistropy of the critical current as a function of the orientation of the field [8]. Brandt analyzed the equation of motion for the sheet current in square and rectangular films in a time-dependent transverse magnetic field. He confirmed the cushion-like flux penetration and the discontinuity lines in the current flow [9]. On the other hand, a very important consequence to consider for a 3D superconducting sample in a tilted magnetic field is the demagnetization effect. There are many experimental and theoretical studies for 3D-systems. Kapolka et al. developed numerical models of the electromagnetic behavior of superconducting rectangular-based bulks and tape stacks. These studies aim at providing a comparison and validation of different theoretical approaches for numerical modeling of high-temperature superconductors. Also, they present numerical modeling and experiments for a cubic Gd Ba Cu O bulk superconductor sample magnetized by means of field cooling [10–13]. The Ginzburg–Landau model has been proven to give a good account of the superconducting properties in samples with several geometries and boundary conditions [14–26]. In the present paper, we investigate the effects of the boundary conditions on the magnetic response of a superconducting parallelepiped immersed in a tilted magnetic field. We consider a superconductor with three types of interfaces: superconductor-vacuum, superconductor-metal, and
Corresponding author. E-mail addresses: [email protected] (J. Barba-Ortega), [email protected] (M.R. Joya), [email protected] (E. Sardella).
https://doi.org/10.1016/j.physc.2019.01.002 Received 16 September 2018; Received in revised form 2 December 2018; Accepted 7 January 2019 Available online 09 January 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved.
Physica C: Superconductivity and its applications 558 (2019) 1–6
J. Barba-Ortega et al.
superconductor-superconductor at a high critical temperature. The paper is structured as follows: In Section 2, we write the dimensionless time-dependent Ginzburg–Landau (TDGL) equations with generalized boundary conditions. In Section 3, we present and discuss the results of the numerical simulations for certain parameters of a superconducting 3D sample for several values of the de Gennes parameter b (or γ, which unifies all types of boundary conditions). In Section 4, we present our conclusions. 2. Theoretical formalism The geometry of the problem that we investigate is illustrated in Fig. 1. The domain Ωsc is filled by the mesoscopic superconducting parallelepiped of thickness c and lateral sizes a and b. The interface between this region and the vacuum is denoted by ∂Ωsc. Because of the demagnetization effects, we need to consider a larger domain Ω of dimensions A × B × C, such that Ωsc ⊂ Ω. We consider a mesoscopic superconducting parallelepiped in the presence of a uniform tilted magnetic field H oriented at an angle of θ with respect to the z axis in the zy plane. The domain Ω is taken as large enough so that the local magnetic field equals the applied field H at the surface ∂Ω. The general form of TDGL equations in dimensionless units are given by:
=
t A t
=
A) 2 +
( i 2
Js 2
(1
| |2 ) , in
× × A , in × A , in
×
,
(1)
Fig. 1. (a) Schematic view of the geometry of the system under investigation. (b) Layout of the studied sample; the external applied magnetic field H is pointed with a θ angle respect to z-axis in the xz plane.
,
sc sc
sc
,
(2)
3. Results and discussion
where
Js = Re[ ¯ ( i
In order to solve the 3D-TDGL equations, the size of the simulation box Ω was taken as ,1 and the inside superconductor has the size a = b = 8 , c = 10 . We used = 1.0, T = 0, a mesh x = y = z = 0.1 , = 0, /6, /4, /3, 0, and with grid size = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.02, 1.04 . In Fig. 2, we present the magnetization in the z axis as a function of the applied tilted magnetic field H for a 3D cube for a) = 0.0, 0.6, 0.8, 1.04, = /4, b) = 1.0, = /4, and d) Mz ( = 0) and = 0, /6, /4, /3, c) = 1.04, Mx ( = /2) . These results show a typical magnetization profile of a mesoscopic superconductor. They exhibit a series of jumps, in which each discontinuity signals the entrance of vortices into the sample. Notice that the number of jumps and the transition fields vary significantly with γ [21]. We see that the slope of the magnetization curves show differences in the Meissner state when θ and γ vary. The sample becomes more (less) diamagnetic with increasing γ (decreasing θ). The sample is less diamagnetic when it is in contact with a ferromagnetic material = 0 . We found an exponential (linear) behavior of the maximum of the magnetization 4 MMax as a function of γ and θ. The 4 MMax 0.09 + 0.002 exp( /0.205) for fitted curves correspond to = /4, and 4 MMax 0.459 0.449 / for = 1.0 . Thus the cube seems to be much more diamagnetic when a superconducting interface at a higher critical temperature is chosen (see Fig. 3). We found that first vortex penetration field is H1 ≃ 0.628, for = /6, /4, /3, with a N to N + 1 vortex transition. The superconducting-normal transition field H2 depends strongly on γ (see Fig. 2(a)) and is practically independent of θ (see Fig. 2(b)). For a superconductor-superconductor = /4 for interface at higher critical temperature, = 1.04, and H ∼ 0.25. In the down branch of the magnetic field, we can see a paramagnetic behavior, with N = 1 quantum flux trapped in the sample at (see Fig. 2(c)). Finally, in Fig. 2(d),) we plot the magnetization of the sample for the cases = 0, that is with the magnetic field parallel to the z axis H =0 Mx = 0 ), = /2, (H‖ ≠ 0⇒Mz ≠ 0 and and
(3)
A) ]
is the superconducting current density. In Eqs. (1)–(3), dimensionless units were introduced as follows: The order parameter ψ is in units of = / , the order parameter at the Meissner state, where α and β are two phenomenological constants; the lengths are in units of the coherence length ξ; time is in units of Ginzburg–Landau characteristic /8KB Tc ; fields are in units of Hc2, where Hc2 is the bulk time tGL = second critical field; the vector potential A is in units of ξHc2; and = / is the Ginzburg–Landau parameter. The phase diagram of a mesoscopic superconductor is strongly influenced by the boundary conditions for the order parameter. In general, they are given by the de Gennes boundary conditions:
n ·(i
+ A)
i b
=
×A = H,
at
, ,
at
sc
,
(4) (5)
where n is the normal outer unit vector to the superconductor-medium interface, and b is the de Gennes surface extrapolation length, which describes this medium [27–32]. We have previously stated that Ω is a domain large enough so that the local magnetic field h = × A equals the external applied magnetic field H. It must be emphasized that the space between the interfaces ∂Ωsc and ∂Ω is not filled by any material. The superconductor is covered by a very thin layer of another material, which is contained in the domain Ωsc. This region is described by the de Gennes extrapolation length b. In order to solve equations (1)–(3) numerically, we used the link-variable method as sketched in references [33,34]. We unify the boundary conditions by introducing the para/ b , where δ is the resolution of the mesh grid used to meter = 1 solve Eqs. (1) and (2) numerically [35–37]. This notation allows us to obtain a more comprehensive analysis of the results. Thus (i) 0 < γ < 1 simulates a superconductor-metal (SC-M) interface (b > δ); (ii) = 1 simulates a superconductor-dielectric (SC-D) interface (b → ∞); and (iii) a superconductor-superconductor (SC-SC) interface is described by γ > 1, (b < 0) [14,21].
1
2
Justification for this choice of parameters can be found in reference [22].
Physica C: Superconductivity and its applications 558 (2019) 1–6
J. Barba-Ortega et al.
Fig. 3. Maximum of the magnetization as a function of the γ and θ parameters: theoretical results and exponential fit.
(H = 0 Mz = 0 and H⊥ ≠ 0⇒Mx ≠ 0). We found that H1 = 0.746 for = 0 and H1 = 0.874 for = /2, and that H2 is practically independent of θ. In Fig. 4, we plot (a) the bi-dimensional density contour plot of |ψ|2 in the plane xy in a stationary state and (b) the phase of the order parameter ΔΦ for the boundary conditions with = 1.0, angle = 0 at H = 0.85, 1.10, N = 2, 4, 6, 8 in the up and down branch of the magnetic field (green arrows). As is well known, the phase of the order parameter determines the vorticity in a given region by determining its variation in a closed path around this region. If the vorticity in this region is N, then the phase changes by 2πN. The multi-vortex state is indistinguishable in the density plot of the magnitude of the order parameter and in the magnetic field induction, since they are so tightly packed in a small region. However, the core center is not coincident. In the Figs. 5 and 6, we depict |ψ|2 and magnetic induction B at z = 0 plane for several stationary states. We choose different values of the γ parameter to illustrate the role played by the different boundary conditions on the vortex matter. We take for the order parameter = /4, ( = 1.04, = /4 ), b) (magnetic induction) (a) = 1.0, = 1.04, = /4, ( = 1.0, = /4 ), and c) = 1.0, = /6, ( = 1.0, = /3), at H = 0.800, 0.85, 0.956, 1.00, 1.10 . We see that the vortex configuration (as magnetic induction) are strongly influenced by θ and γ, it shown that is possible to obtain a multi-vortex configuration for
Fig. 2. Magnetization 4πMz as a function of H for a) = 0.0, 0.6, 0.8, 1.04, = /4, b) = 1.0, = 0, /6, /4, /3, c) = 1.04, = /4, d) Mz ( = 0) and Mx ( = /2) .
3
Physica C: Superconductivity and its applications 558 (2019) 1–6
J. Barba-Ortega et al.
Fig. 5. Bi-dimensional density contour plot of |ψ|2 in the plane xy, for the boundary conditions with (a) = 1.0, angle = /4, b) = 1.04, angle = /4, = /6, in the up and downbranch of = 1.0, c) angle H = 0.800, 0.85, 0.956, 1.00, 1.10 (green arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
increasing the magnetic field, more vortices enter the sample and form a triangular configuration at H = 1.0 and H = 0.956 for = /4 and = 1.0, 1.04, respectively (in the up branch of the magnetic field). In Fig. 5(c), the triangular configuration is found at a low magnetic field H = 0.85 for = /6 . In Fig. 7, we plotted the density profile of |ψ|2 in = 0, /6, /4, and a) H = 0.800, b) = 1.0, the plane xz for H = 0.850, and c) H = 1.00 . It can easily be seen that we obtain the profile of the magnetic fluxoid parallel to H for = 0, but as is expected, the central core grows in size at higher magnetic fields. For = /4, /6, we have a different magnetic signature. As can be seen, now we have a significant variation of the order parameter throughout the thickness of the cube. Notice that for the case = /6, /4 has three and two tilted cores, respectively. Now it seems that the central core grows in size as the external magnetic field increases. At a high magnetic field, on increasing θ the vortex profiles are interlaced, forming two larger cores. So the tilted profile of the vortex is due to the nonsymmetrical super-currents in the sample as a consequence of the tilted external magnetic field.
Fig. 4. (a) Bi-dimensional density contour plot of |ψ|2 in the plane xy and (b) phase of the order parameter ΔΦ, for the boundary conditions with = 1.0, angle = 0 at H = 0.85, 1.10, N = 2, 4, 6, 8 in the up and downbranch of the magnetic field (green arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
low vorticity N = 2, 4 . In Figs. 5 and 6, we depict |ψ|2 and magnetic induction B in the z = 0 plane for several stationary states. We choose different values of the γ parameter to illustrate the effect of the different boundary conditions on the vortex matter. For the order parameter (magnetic induction), we take (a) = 1.0, = /4, ( = 1.04, = /4 ), = /4 ), and c) = 1.0, = /4, ( = 1.0, = /6, b) = 1.04, ( = 1.0, = /3), at H = 0.800, 0.85, 0.956, 1.00, 1.10 . We see that the vortex configuration (as magnetic induction) is strongly influenced by θ and γ. It shown that it is possible to obtain a multi-vortex configuration for low vorticity N = 2, 4 . As we can see, in Fig. 5(a) and (b), N = 2 vortices are located in the sample asymmetrically at H = 0.8. On
4. Conclusions We have shown that a tilted magnetic field and different surface conditions produce different signatures in the vortex configurations and magnetization curves of a small 3D superconducting cube in the presence of an external applied magnetic field. Nucleation and full expulsion of vortices, magnetic fields, the vortex configurations, and the
4
Physica C: Superconductivity and its applications 558 (2019) 1–6
J. Barba-Ortega et al.
Fig. 6. Magnetic induction B at z = 0 plane, for the boundary conditions with (a) H = 0.800, 0.85, 0.956, 1.00 .
= 1.04,
= /4, b)
= 1.0,
= /4, c)
= 1.0,
= /3, for
tilted magnetic field with = /4 and a function of the angle of incidence of the magnetic field 4 MMax 0.459 0.449 / for a superconductor-dielectric interface. References
Fig. 7. Bi-dimensional density profile of |ψ|2 in the plane xz, for = 0, /6, /4, and a) H = 0.800 b) H = 0.850, c) H = 1.00 .
[1] V.G. Kogan, Phys. Rev. B 24 (1981) 1572. [2] V.G. Kogan, Phys. Rev. B 38 (1988) 7049. [3] D.E. Farrel, C.M. William, S.A. Wolf, N.P. Bansal, V.G. Kogan, Phys. Rev. Lett. 61 (1988) 2805. [4] D. Feinberg, C. Villard, Mod. Rev. Lett. B 4 (9) (1990). [5] A. Buzdin, Y. Simonov, Physica C 175 (1991) 143. [6] I.V. Grigorieva, L.A. Gurevich, L.Y. Vinnikov, Physica C 195 (1992) 327. [7] I.V. Grigorieva, J.W. Steeds, K. Sasaki, Phys. Rev. B48 (1993) 16865. [8] B.I. Ivlev, Y.N. Ovchinnikov, V.L. Pokrovsky, Europhys. Lett. 3 (1990) 187. [9] E.H. Brandt, Phys. Rev. Lett. 74 (1995) 3025. [10] E. Pardo, M. Kapolka, Supercond. Sci. Technol. 30 (2017) 064007. [11] E. Pardo, M. Kapolka, J. Comp. Phys. 344 (2017) 339. [12] M. Kapolka, V.M.R. Zermeno, S. Zou, A. Morandi, P.L. Ribani, E. Pardo, F. Grilli, IEEE Trans. Appl. Supercond. 28 (2018) 8201206. [13] M. Kapolka, J. Srpcic, D. Zhou, M.D. Ainslie, E. Pardo, A.R. Dennis, IEEE Trans. Appl. Supercond. 28 (2018) 6801495. [14] J. Barba-Ortega, E. Sardella, R. Zadorosny, Phys. Lett. A 382 (2018) 215. [15] R.I. Rey, A.R. Alvarez, C. Carballeira, J. Mosqueira, F. Vidal, S. Salem, A.D. Alvarenga, R. Zhang, H. Luo, Supercond. Sci. Technol. 27 (2014) 075001. [16] H. Suderow, I. Guillamon, J.G. Rodrigo, S. Vieira1, Supercond. Sci. Technol. 27 (2014) 063001. [17] D. Glotov, Z. Angew, Math. Phys. 62 (2011) 891. [18] S.J. Chapman, Q. Du, M.D. Gunzburger, Z. Angew, Math. Phys. 47 (1993) 410. [19] G.R. Berdiyorov, M.M. Doria, A.R. de C. Romaguera, M.V. Milosevic, E.H. Brandt, F.M. Peeters, Phys. Rev. B87 (2013) 184508. [20] L.F. Zhang, L. Covaci, M.V. Milosevic, G.R. Berdiyorov, F.M. Peeters, Phys. Rev. Lett. 109 (2012) 107001. [21] J. Barba-Ortega, E. Sardella, J.A. Aguiar, Supercond. Sci. Technol. 24 (2011) 015001. [22] J. Barba-Ortega, E. Sardella, J.A. Aguiar, Phys. Lett. A 379 (2015) 732.
= 1.0,
number of vortices can be strongly influenced by the anisotropic surface conditions and the angle of incidence of the magnetic field. We found an analytical behavior of the maximum diamagnetism as a function of 4 MMax 0.09 + 0.002 exp( /0.205) for a the boundary condition
5
Physica C: Superconductivity and its applications 558 (2019) 1–6
J. Barba-Ortega et al. [23] Y. Chen, M.M. Doria, F.M. Peeters, Phys. Rev. B77 (2008) 054511. [24] P.J. Pereira, V.V. Moshchalkov, L.F. Chibotaru, J. Phys. Conf. Ser. 490 (2014) 01222. [25] B. Xu, M.V. Milosevic, F.M. Peeters, Phys. Rev. B77 (2008) 144509. [26] M.M. Doria, R.M. Romaguera, F.M. Peeters, Phys. Rev. B75 (2007) 064505. [27] P.G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, New York, 1994. [28] P.G. de Gennes, J. Matricon, Rev. Mod. Phys. 36 (1964) 45. [29] E.A. Andrushin, V.L. Ginzburg, A.P. Silin, Usp. Fiz. Nauk. 163 (1997) 105. [30] R.O. Zaitsev, Zh. Eksp. Teor. Fiz. 48 (1965) 1759. [31] J. Simonin, Phys. Rev. B33 (1986) 7830.
[32] H.J. Fink, S.B. Haley, C.V. Giuraniuc, V.F. Kozhevnikov, J.O. Indekeu, Mol. Phys. 103 (2005) 21. [33] W.D. Gropp, H.G. Kaper, G.K. Leaf, D.M. Levine, M. Palumbo, V.M. Vinokur, J. Comput. Phys 123 (1996) 254. [34] G. Buscaglia, C. Bolech, C. Lopez, Connectivity and Superconductivity, Springer, Heidelberg, 2000. [35] G. Piacente, F.M. Peeters, Phys. Rev B72 (2005) 205208. [36] R.M. da Silva, M.V. Milosevic, A.A. Shanenko, F.M. Peeters, J.A. Aguiar, Sci. Rep. 5 (2015) 12695. [37] A.A. Shanenko, J.A. Aguiar, A. Vagov, M.D. Croitoru, M.V. Milosevic, Supercond. Sci. Technol 28 (2015) 054001.
6
Contents lists available at ScienceDirect
Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc
Superconducting 3D-sample with general boundary conditions in a tilted magnetic field
T
J. Barba-Ortega ,a, M.R. Joyaa, E. Sardellab ⁎
a b
Departamento de Física, Universidad Nacional de Colombia, Bogotá, Colombia Departamento de Física, Universidade Estadual Paulista, UNESP, Baurú, SP, Brazil
ARTICLE INFO
ABSTRACT
Keywords: Vortex Ginzburg–Landau Magnetization Mesoscopics Superconductor Tilded field
We consider a mesoscopic superconducting three-dimensional cube immersed in a tilted magnetic field H oriented along an angle θ with respect to the z-axis, in the xz plane. The sample is in contact with a metallic material or with another superconductor in the xz and yz planes, while the z = 0 plane remains in contact with a dielectric material. These boundary conditions are simulated by the de Gennes extrapolation length b. We analyzed the magnetic induction, the superconducting electron density, and the magnetization curves as functions of H for different values of b on the lateral surfaces of the sample. We show that the magnetization and the vortex configuration depend on θ and b. An analytical linear and exponential dependence of the maximum of the magnetization on θ and b, respectively, was found.
PACS: 74.20.De 74.25.Bt 74.25.Uv 74.25.Wx
1. Introduction As is well known, in spatially homogeneous anisotropic superconducting samples, the Shubnikov vortex lattice consists of rectilinear magnetic quantum flux aligned along the external applied magnetic field. The magnetic energy depends on the orientation of the magnetic field with respect to the crystallographic axis [1–3]. Several theoretical and experimental studies of superconductors systems in the presence of magnetic fields have been performed in the last few decades. For example, Feinberg et al. assumed that the vortices in this magnetic field have a kink structure and the first critical magnetic field depends on the angle [4]. Buzdin et al. studied the angular and field dependences of the magnetization of superconductors in tilted magnetic fields. They discussed the inclined Abrikosov vortex lattice in quasi one- and two-dimensional superconductors and found that in contrast to the typical situation with one maximum of the magnetization at the first vortex entry magnetic field, an additional maximum appears in the region of the higher field [5]. Grigorieva et al., observed the vortex structure experimentally using the Bitter technique in YBaCuO and YBa2 (Cu1 x Alx )3 O7 single crystals. They found that the vortex configuration remained unchanged for an interval of tilt angles θ > 25 with respect to the c-axis [6,7]. Ivlev et al. showed that at high Tc, superconductors with a well-pronounced structure of the order parameter, a
⁎
magnetic field tilted to the superconducting planes gives rise to a system of kinks on the vortex lines. This leads to strong anistropy of the critical current as a function of the orientation of the field [8]. Brandt analyzed the equation of motion for the sheet current in square and rectangular films in a time-dependent transverse magnetic field. He confirmed the cushion-like flux penetration and the discontinuity lines in the current flow [9]. On the other hand, a very important consequence to consider for a 3D superconducting sample in a tilted magnetic field is the demagnetization effect. There are many experimental and theoretical studies for 3D-systems. Kapolka et al. developed numerical models of the electromagnetic behavior of superconducting rectangular-based bulks and tape stacks. These studies aim at providing a comparison and validation of different theoretical approaches for numerical modeling of high-temperature superconductors. Also, they present numerical modeling and experiments for a cubic Gd Ba Cu O bulk superconductor sample magnetized by means of field cooling [10–13]. The Ginzburg–Landau model has been proven to give a good account of the superconducting properties in samples with several geometries and boundary conditions [14–26]. In the present paper, we investigate the effects of the boundary conditions on the magnetic response of a superconducting parallelepiped immersed in a tilted magnetic field. We consider a superconductor with three types of interfaces: superconductor-vacuum, superconductor-metal, and
Corresponding author. E-mail addresses: [email protected] (J. Barba-Ortega), [email protected] (M.R. Joya), [email protected] (E. Sardella).
https://doi.org/10.1016/j.physc.2019.01.002 Received 16 September 2018; Received in revised form 2 December 2018; Accepted 7 January 2019 Available online 09 January 2019 0921-4534/ © 2019 Elsevier B.V. All rights reserved.
Physica C: Superconductivity and its applications 558 (2019) 1–6
J. Barba-Ortega et al.
superconductor-superconductor at a high critical temperature. The paper is structured as follows: In Section 2, we write the dimensionless time-dependent Ginzburg–Landau (TDGL) equations with generalized boundary conditions. In Section 3, we present and discuss the results of the numerical simulations for certain parameters of a superconducting 3D sample for several values of the de Gennes parameter b (or γ, which unifies all types of boundary conditions). In Section 4, we present our conclusions. 2. Theoretical formalism The geometry of the problem that we investigate is illustrated in Fig. 1. The domain Ωsc is filled by the mesoscopic superconducting parallelepiped of thickness c and lateral sizes a and b. The interface between this region and the vacuum is denoted by ∂Ωsc. Because of the demagnetization effects, we need to consider a larger domain Ω of dimensions A × B × C, such that Ωsc ⊂ Ω. We consider a mesoscopic superconducting parallelepiped in the presence of a uniform tilted magnetic field H oriented at an angle of θ with respect to the z axis in the zy plane. The domain Ω is taken as large enough so that the local magnetic field equals the applied field H at the surface ∂Ω. The general form of TDGL equations in dimensionless units are given by:
=
t A t
=
A) 2 +
( i 2
Js 2
(1
| |2 ) , in
× × A , in × A , in
×
,
(1)
Fig. 1. (a) Schematic view of the geometry of the system under investigation. (b) Layout of the studied sample; the external applied magnetic field H is pointed with a θ angle respect to z-axis in the xz plane.
,
sc sc
sc
,
(2)
3. Results and discussion
where
Js = Re[ ¯ ( i
In order to solve the 3D-TDGL equations, the size of the simulation box Ω was taken as ,1 and the inside superconductor has the size a = b = 8 , c = 10 . We used = 1.0, T = 0, a mesh x = y = z = 0.1 , = 0, /6, /4, /3, 0, and with grid size = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.02, 1.04 . In Fig. 2, we present the magnetization in the z axis as a function of the applied tilted magnetic field H for a 3D cube for a) = 0.0, 0.6, 0.8, 1.04, = /4, b) = 1.0, = /4, and d) Mz ( = 0) and = 0, /6, /4, /3, c) = 1.04, Mx ( = /2) . These results show a typical magnetization profile of a mesoscopic superconductor. They exhibit a series of jumps, in which each discontinuity signals the entrance of vortices into the sample. Notice that the number of jumps and the transition fields vary significantly with γ [21]. We see that the slope of the magnetization curves show differences in the Meissner state when θ and γ vary. The sample becomes more (less) diamagnetic with increasing γ (decreasing θ). The sample is less diamagnetic when it is in contact with a ferromagnetic material = 0 . We found an exponential (linear) behavior of the maximum of the magnetization 4 MMax as a function of γ and θ. The 4 MMax 0.09 + 0.002 exp( /0.205) for fitted curves correspond to = /4, and 4 MMax 0.459 0.449 / for = 1.0 . Thus the cube seems to be much more diamagnetic when a superconducting interface at a higher critical temperature is chosen (see Fig. 3). We found that first vortex penetration field is H1 ≃ 0.628, for = /6, /4, /3, with a N to N + 1 vortex transition. The superconducting-normal transition field H2 depends strongly on γ (see Fig. 2(a)) and is practically independent of θ (see Fig. 2(b)). For a superconductor-superconductor = /4 for interface at higher critical temperature, = 1.04, and H ∼ 0.25. In the down branch of the magnetic field, we can see a paramagnetic behavior, with N = 1 quantum flux trapped in the sample at (see Fig. 2(c)). Finally, in Fig. 2(d),) we plot the magnetization of the sample for the cases = 0, that is with the magnetic field parallel to the z axis H =0 Mx = 0 ), = /2, (H‖ ≠ 0⇒Mz ≠ 0 and and
(3)
A) ]
is the superconducting current density. In Eqs. (1)–(3), dimensionless units were introduced as follows: The order parameter ψ is in units of = / , the order parameter at the Meissner state, where α and β are two phenomenological constants; the lengths are in units of the coherence length ξ; time is in units of Ginzburg–Landau characteristic /8KB Tc ; fields are in units of Hc2, where Hc2 is the bulk time tGL = second critical field; the vector potential A is in units of ξHc2; and = / is the Ginzburg–Landau parameter. The phase diagram of a mesoscopic superconductor is strongly influenced by the boundary conditions for the order parameter. In general, they are given by the de Gennes boundary conditions:
n ·(i
+ A)
i b
=
×A = H,
at
, ,
at
sc
,
(4) (5)
where n is the normal outer unit vector to the superconductor-medium interface, and b is the de Gennes surface extrapolation length, which describes this medium [27–32]. We have previously stated that Ω is a domain large enough so that the local magnetic field h = × A equals the external applied magnetic field H. It must be emphasized that the space between the interfaces ∂Ωsc and ∂Ω is not filled by any material. The superconductor is covered by a very thin layer of another material, which is contained in the domain Ωsc. This region is described by the de Gennes extrapolation length b. In order to solve equations (1)–(3) numerically, we used the link-variable method as sketched in references [33,34]. We unify the boundary conditions by introducing the para/ b , where δ is the resolution of the mesh grid used to meter = 1 solve Eqs. (1) and (2) numerically [35–37]. This notation allows us to obtain a more comprehensive analysis of the results. Thus (i) 0 < γ < 1 simulates a superconductor-metal (SC-M) interface (b > δ); (ii) = 1 simulates a superconductor-dielectric (SC-D) interface (b → ∞); and (iii) a superconductor-superconductor (SC-SC) interface is described by γ > 1, (b < 0) [14,21].
1
2
Justification for this choice of parameters can be found in reference [22].
Physica C: Superconductivity and its applications 558 (2019) 1–6
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Fig. 3. Maximum of the magnetization as a function of the γ and θ parameters: theoretical results and exponential fit.
(H = 0 Mz = 0 and H⊥ ≠ 0⇒Mx ≠ 0). We found that H1 = 0.746 for = 0 and H1 = 0.874 for = /2, and that H2 is practically independent of θ. In Fig. 4, we plot (a) the bi-dimensional density contour plot of |ψ|2 in the plane xy in a stationary state and (b) the phase of the order parameter ΔΦ for the boundary conditions with = 1.0, angle = 0 at H = 0.85, 1.10, N = 2, 4, 6, 8 in the up and down branch of the magnetic field (green arrows). As is well known, the phase of the order parameter determines the vorticity in a given region by determining its variation in a closed path around this region. If the vorticity in this region is N, then the phase changes by 2πN. The multi-vortex state is indistinguishable in the density plot of the magnitude of the order parameter and in the magnetic field induction, since they are so tightly packed in a small region. However, the core center is not coincident. In the Figs. 5 and 6, we depict |ψ|2 and magnetic induction B at z = 0 plane for several stationary states. We choose different values of the γ parameter to illustrate the role played by the different boundary conditions on the vortex matter. We take for the order parameter = /4, ( = 1.04, = /4 ), b) (magnetic induction) (a) = 1.0, = 1.04, = /4, ( = 1.0, = /4 ), and c) = 1.0, = /6, ( = 1.0, = /3), at H = 0.800, 0.85, 0.956, 1.00, 1.10 . We see that the vortex configuration (as magnetic induction) are strongly influenced by θ and γ, it shown that is possible to obtain a multi-vortex configuration for
Fig. 2. Magnetization 4πMz as a function of H for a) = 0.0, 0.6, 0.8, 1.04, = /4, b) = 1.0, = 0, /6, /4, /3, c) = 1.04, = /4, d) Mz ( = 0) and Mx ( = /2) .
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Physica C: Superconductivity and its applications 558 (2019) 1–6
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Fig. 5. Bi-dimensional density contour plot of |ψ|2 in the plane xy, for the boundary conditions with (a) = 1.0, angle = /4, b) = 1.04, angle = /4, = /6, in the up and downbranch of = 1.0, c) angle H = 0.800, 0.85, 0.956, 1.00, 1.10 (green arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
increasing the magnetic field, more vortices enter the sample and form a triangular configuration at H = 1.0 and H = 0.956 for = /4 and = 1.0, 1.04, respectively (in the up branch of the magnetic field). In Fig. 5(c), the triangular configuration is found at a low magnetic field H = 0.85 for = /6 . In Fig. 7, we plotted the density profile of |ψ|2 in = 0, /6, /4, and a) H = 0.800, b) = 1.0, the plane xz for H = 0.850, and c) H = 1.00 . It can easily be seen that we obtain the profile of the magnetic fluxoid parallel to H for = 0, but as is expected, the central core grows in size at higher magnetic fields. For = /4, /6, we have a different magnetic signature. As can be seen, now we have a significant variation of the order parameter throughout the thickness of the cube. Notice that for the case = /6, /4 has three and two tilted cores, respectively. Now it seems that the central core grows in size as the external magnetic field increases. At a high magnetic field, on increasing θ the vortex profiles are interlaced, forming two larger cores. So the tilted profile of the vortex is due to the nonsymmetrical super-currents in the sample as a consequence of the tilted external magnetic field.
Fig. 4. (a) Bi-dimensional density contour plot of |ψ|2 in the plane xy and (b) phase of the order parameter ΔΦ, for the boundary conditions with = 1.0, angle = 0 at H = 0.85, 1.10, N = 2, 4, 6, 8 in the up and downbranch of the magnetic field (green arrows). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
low vorticity N = 2, 4 . In Figs. 5 and 6, we depict |ψ|2 and magnetic induction B in the z = 0 plane for several stationary states. We choose different values of the γ parameter to illustrate the effect of the different boundary conditions on the vortex matter. For the order parameter (magnetic induction), we take (a) = 1.0, = /4, ( = 1.04, = /4 ), = /4 ), and c) = 1.0, = /4, ( = 1.0, = /6, b) = 1.04, ( = 1.0, = /3), at H = 0.800, 0.85, 0.956, 1.00, 1.10 . We see that the vortex configuration (as magnetic induction) is strongly influenced by θ and γ. It shown that it is possible to obtain a multi-vortex configuration for low vorticity N = 2, 4 . As we can see, in Fig. 5(a) and (b), N = 2 vortices are located in the sample asymmetrically at H = 0.8. On
4. Conclusions We have shown that a tilted magnetic field and different surface conditions produce different signatures in the vortex configurations and magnetization curves of a small 3D superconducting cube in the presence of an external applied magnetic field. Nucleation and full expulsion of vortices, magnetic fields, the vortex configurations, and the
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Fig. 6. Magnetic induction B at z = 0 plane, for the boundary conditions with (a) H = 0.800, 0.85, 0.956, 1.00 .
= 1.04,
= /4, b)
= 1.0,
= /4, c)
= 1.0,
= /3, for
tilted magnetic field with = /4 and a function of the angle of incidence of the magnetic field 4 MMax 0.459 0.449 / for a superconductor-dielectric interface. References
Fig. 7. Bi-dimensional density profile of |ψ|2 in the plane xz, for = 0, /6, /4, and a) H = 0.800 b) H = 0.850, c) H = 1.00 .
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