Giant vortex state in a mesoscopic superconducting thin ring

doi:10.1016/j.physc.2021.1353917

Physica C: Superconductivity and its Applications

Volume 588, 15 September 2021, 1353917

https://doi.org/10.1016/j.physc.2021.1353917 Get rights and content

Highlights

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The effect of the size of the thin ring on the superconducting vortex state is given.
•
The vortex characteristics of a thin superconducting ring are different from those of a thin superconducting disk.
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The mesoscopic rings with inner radius and outer radius between and contains the two vortex state, only Meissner State and giant vortex state.
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For a circular ring with a certain outer radius, the upper nucleation field of a vortex state with a certain vorticity has a peak value with the increase of the inner radius.

Abstract

Using numerical treatment of the Ginzburg–Landau (GL) theory, the paper probes into superconducting thin rings. Systematic analysis is conducted for the free energy of the thin rings with different inner radii and outer radii of the ring. The effect of inner radii and outer radii of the ring on the vortex state are discussed. Furthermore, we obtain a phase diagram, where the giant vortex state regimes are determined by geometric parameters. The characteristics of the ring different from the disk are obtained.

Graphical abstract

Introduction

Due to the progress of modern nanofabrication technology, the study of mesoscopic samples has become a frontier field of condensed matter physics since Van Kampen founded mesoscopic in 1981 [1], [2], [3], [4], [5], [6]. A mesoscopic sample is such a sample that its size is comparable to the coherence length ( $λ$ ) and the magnetic-field penetration length ( $ξ$ ). Because of the particularity of size, mesoscopic superconductors exhibit many different properties of macroscopic superconductors [7], [8], [9], [10], [11], [12]. The existence of giant vortex states and multi-vortex states has been observed experimentally on the aluminum disk by the multiple-small-tunnel-junction method [13]. When the extent of the mesoscopic superconductor is equal to the coherence length or penetration depth of the superconducting characteristic size, the ground state of the superconducting system mainly exists in the form of the Meissner state or the giant vortex state [14], [15], [16], [17] (The vortex state with zero angular momentum quantum number is also called Meissner state). Only when the size of the disk or ring is relatively large (such as several times of the coherent length), can the multivortex state appear in the thin disk or ring [18], [19], [20]. Moreover, the stable states are mostly in the form of giant vortex states, and the multi vortex states are mostly metastable states [21], [22], [23]. The prominent feature of the wave function of these giant states is that the wave function exhibits an axisymmetric distribution and has a fixed angular momentum quantum number, also known as the vorticity number. The different oscillation curves of the $T_{c} (H)$ behavior in low flux and high flux regime given in the reference [24] are essentially the change between the giant vortex states and the normal state. Baelus et al. have also studied the giant vortex states of the disk with holes, and have obtained the phase diagrams of the upper nucleation field of each giant vortex state varying with the size of hole [19]. Therefore, it is very important to study giant vortex states in mesoscopic superconductors. What is the relationship between the existence of these giant vortex states or Meissner State and the size of mesoscopic superconducting samples? In this paper, a large number of mesoscopic superconductors thin rings with different sizes are simulated to reveal the influence of the mesoscopic superconductor size on the state of the Meissner state or the giant vortex states under the action of a uniform magnetic field. Our results provide a theoretical reference for the application of mesoscopic superconductors.

Section snippets

Theoretical model

According to Ginzburg–Landau theory, the free energy of superconducting systems is $F = \frac{2}{V} {\int d V {- | ψ |}^{2} + {| ψ |}^{4} + | (\vec{\nabla} - i \vec{A}) {ψ) |}^{2}}$

We measure the distance in units of coherence length $ξ = ℏ / \sqrt{2 m [- α (T)]}$ ,the vector potential $\vec{A}$ in $c ℏ / 2 e ξ (T)$ , the order parameter $ψ$ in $ψ_{0} = \sqrt{- α (T) / β}$ , and the free energy $F$ in $F_{0} = α {(T)}^{2} / 2 β$ , the magnetic field $H$ in $H_{c 2} = c ℏ / 2 e {ξ (T)}^{2} = κ \sqrt{2} H_{c}$ , where $H_{c}$ is the thermodynamical critical field. The magnetic shielding effect is not considered here.

The system we studied is a mesoscopic superconducting ring

The free energy of giant vortex

The giant vortex state has cylindrical symmetry and consequently the order parameter can be written as formula (9). The giant vortex states are labelled with their $L$ value. To understand the superconductivity of thin rings, we calculate the free energy versus the magnetic field of the giant vortex for various ring dimensions from Eq. (12). In Fig. 2, the free energy curves are given as a function of the applied magnetic field for a superconducting ring with the same inner radius $R_{i} / ξ = 0.5$ and

The phase diagram of giant vortex states

First, with a fixed inner radius $R_{i} / ξ = 0.5$ and outer radius from $R_{o} / ξ$ from 0.5 to 7.0, the free energy of each thin rings is calculated. The nucleation field of each giant vortex state for each ring can be obtained. The nucleation field dependence of the outer radius $R_{o} / ξ$ is shown in the Fig. 3. The abscissa coordinate is the outer radius of the mesoscopic superconducting ring, and the longitudinal coordinate is the nucleation field of each giant vortex state. Different colors represent the

Conclusion

Taking the inner radius $R_{i} / ξ = 0.5$ and the outer radius $R_{o} / ξ = 6.0$ fixed respectively as examples, the free energy of different outer radius or inner radius of mesoscopic thin ring is studied by Ginzburg–Landau theory. We have found the relationship between the nucleation magnetic fields and the size of the mesoscopic ring. The mesoscopic rings with inner radius $R_{i} / ξ = 0.5$ and outer radius $R_{o} / ξ$ between 1 and 1.5 only contains two vortex states, only $L = 0$ Meissner State and $L = 1$ giant vortex state. As

CRediT authorship contribution statement

Ji-Xue Liu: Conceptualization, Methodology, Software, Visualization, Investigation. Liang-Ma Shi: Data curation, Writing – original draft. Guo-Qiao Zha: Supervision, Formal analysis, Writing – review & editing.

Declaration of Competing Interest

Authors declare that they have no conflict of interest.

Acknowledgments

This work is supported by the open project of Shanghai Key Laboratory of High Temperature Superconductors.

References (28)

D. Saint-James et al.
Onset of superconductivity in decreasing fields
Phys. Lett.
(1963)
V. Bruyndoncx et al.
Nucleation of superconductivity in a mesoscopic loop of finite width
Phys. C
(2000)
A.K. Geim et al.
Non-quantized penetration of magnetic field in the vortex state of superconductors
Nature
(2000)
Bluhm et al.
Persistent currents in normal metal rings
Phys. Rev. Lett.
(2009)
F. Liang
Electron teleportation via Majorana bound states in a mesoscopic superconductor
Phys. Rev. Lett.
(2010)
A. Nava et al.
Persistent current and zero-energy Majorana modes in a $p$ -wave disordered superconducting ring
Phys. Rev. B
(2017)
A. Camjayi et al.
Fractional spin and Josephson effect in time-reversal-invariant topological superconductors
Phys. Rev. Lett.
(2017)
F.K. de Vries et al.
$h / e$ superconducting quantum interference through trivial edge states in InAs
Phys. Rev. Lett.
(2018)
F.V. Kusmartsev
Destruction of the meissner effect in granular high-temperature superconductors
Phys. Rev. Lett.
(1992)
D. Domnguez et al.
Phenomenological theory of the paramagnetic meissner effect
Phys. Rev. Lett.
(1994)

A.K. Geim et al.

Paramagnetic meissner effect in small superconductors

Nature

(1998)

B.J. Baelus et al.

Vortex shells in mesoscopic superconducting disks

Phys. Rev. B

(2004)

G. Schwiete et al.

Persistent current in small superconducting rings

Phys. Rev. Lett.

(2009)

L.M. Shi et al.

Vortices of mesoscopic rings in an external magnetic field: phenomenological Ginzburg–Landau theory

Phys. Rev. B

(2009)

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View full text

Giant vortex state in a mesoscopic superconducting thin ring

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Theoretical model

The free energy of giant vortex

The phase diagram of giant vortex states

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Phys. Lett.

Phys. C

Non-quantized penetration of magnetic field in the vortex state of superconductors

Nature

Persistent currents in normal metal rings

Phys. Rev. Lett.

Electron teleportation via Majorana bound states in a mesoscopic superconductor

Phys. Rev. Lett.

Persistent current and zero-energy Majorana modes in a p-wave disordered superconducting ring

Phys. Rev. B

Fractional spin and Josephson effect in time-reversal-invariant topological superconductors

Phys. Rev. Lett.

h/e superconducting quantum interference through trivial edge states in InAs

Phys. Rev. Lett.

Destruction of the meissner effect in granular high-temperature superconductors

Phys. Rev. Lett.

Phenomenological theory of the paramagnetic meissner effect

Phys. Rev. Lett.

Paramagnetic meissner effect in small superconductors

Nature

Vortex shells in mesoscopic superconducting disks

Phys. Rev. B

Persistent current in small superconducting rings

Phys. Rev. Lett.

Vortices of mesoscopic rings in an external magnetic field: phenomenological Ginzburg–Landau theory

Phys. Rev. B

Persistent current and zero-energy Majorana modes in a $p$ -wave disordered superconducting ring

$h / e$ superconducting quantum interference through trivial edge states in InAs