Crossover from type I to type II regime of

mesoscopic superconductors of the first group

Leonardo Rodrigues Cadorim1, Thiago de Oliveira Calsolari1,

Rafael Zadorosny2, Edson Sardella2

1Departamento de F́ısica, Faculdade de Ciências, Universidade Estadual Paulista

(UNESP), Caixa Postal 473, 17033-360, Bauru-SP, Brazil
2Departamento de F́ısica e Qúımica, Faculdade de Engenharia de Ilha Solteira,

Universidade Estadual Paulista (UNESP), Caixa Postal 31, 15385-000 Ilha

Solteira-SP, Brazil

E-mail: edson.sardella@unesp.br

July 2019

Abstract. In the present work, we have studied the crossover between type I and type

II superconductivity on mesoscopic superconducting thin films by numerically solving

the 3D Ginzburg-Landau equations. We determined the dependence on temperature

of the critical Ginzburg-Landau parameter κc(d), below which the superconductor

behaves as type I for a given thickness d of the film. The effect of the sample dimensions

on this crossover was also investigated. Additionally, we report a novel giant vortex

configuration with a local minimum of the magnetic field at the centre of the core.

Finally, we present certain results suggesting that the vortex-vortex interaction is not

monotonic, varying instead from long-range attraction to short-range repulsion.

1. Introduction

The distinct magnetic properties of type I and type II bulk superconductors lie in the

signal of the surface energy density, which is positive for the former and negative for

the latter. The transition between these two types are given by the Ginzburg-Landau

parameter κ, the superconductor being of type II when κ > 1/
√

2 and of type I if

κ < 1/
√

2. The former is energetically favourable to the penetration of vortices; in the

latter, the penetration of magnetic flux leads the superconductor to the normal state.

Tinkham [1] first showed, using an argument based on fluxoid quantization, that

superconducting thin films, even of material with κ < 1/
√

2, can behave like type II if the

film is sufficiently thin, possessing a second-order phase transition to the normal state.

Pearl [2] showed that, unlike bulk superconductors, vortices in very thin films of type I

have a long-range repulsion, thus making possible the formation of a vortex lattice. In an

analysis similar to Abrikosov [3], Maki [4] mathematically proved Tinkham’s hypothesis

that a fluxoid structure is indeed a solution to the Ginzburg-Landau equations for a



L. R. Cadorim et al. 2

thin film of a type I superconductor, provided that the film is thinner than a certain

critical thickness.

Later work by Lasher [5] and Callaway [6] extended Maki’s analysis by considering

other types of vortex structures, like the possibility of the formation of giant vortices.

Lasher showed that, although very thin films do exhibit a triangular lattice, films of

intermediate thickness possess a more complex structure, allowing for different patterns,

including vortices with more than a quantum flux. He then predicted the existence of

two critical thicknesses, one corresponding to the triangular lattice phase and a second,

larger one delimiting the mixed and the intermediate states in the film.

Much numerical work has been carried out on these giant vortices configurations

in superconducting thin films. For example, Schweigert et al. [7] studied the vortex

phase diagrams on thin disks of various radii and thicknesses, finding the transition

between the multivortex and giant vortex configurations. Similarly, Shi et al. [8] found

the same transition while studying mesoscopic rings of variable radii. Berdiyorov et al.

[9] analysed the vortex structures in thin films with an antidot array. By varying the

effective Ginzburg-Landau parameter and the periodicity of the antidot lattice, they

provided a structural phase diagram for such a system. Palonen et al. [10] performed

simulations for low κ type I superconductors, finding that giant vortex configurations

are stable in intermediate thickness films, but for sufficiently thin films only single vortex

states are stable. They also speculated that the giant vortex formation is possible due

to a short-range attraction and a long-range repulsion. Córdoba et al. [11] analysed

the stability of the giant vortices, varying both the temperatures and thicknesses of

the film. They found that the formation of those specimens is hindered by increases in

temperature.

Dantas et al. [12] used a set of constrained Ginzburg-Landau equations to calculate

the vortex-vortex interaction potential of a series of thin films of different thicknesses

and κ values. They found that vortex-vortex interaction is non-monotonic; its behaviour

depends on the film’s thickness and κ. They then constructed a phase diagram delimiting

the regions where the film showed a single fluxoid lattice, giant vortex configuration,

and type I behaviour. With this phase diagram, they were able to find the critical κ

value that makes possible the formation of a vortex lattice in thin films. The value they

encountered is different from the κeff = 2κ2/d that is generally assumed for thin films.

The present study investigates the influence of temperature and the surface effects

introduced by the mesoscopic dimensionality of the samples in the critical parameters of

this well-known crossover from type I to type II superconductivity in superconducting

thin films.

By numerically solving the 3D Ginzburg-Landau equations, we studied the

temperature dependence of the crossover between type I and type II regimes for

mesoscopic superconducting thin films. Type I superconducting mesoscopic samples

were simulated, and an expression relating the Ginzburg-Landau parameter of the

material to its critical thickness was obtained, above which the nucleation of vortices

becomes possible for different temperatures. We then studied how the dimensions of the



L. R. Cadorim et al. 3

film that are perpendicular to the applied field influence the previously found expressions

of critical thicknesses. We also analysed the structures of the vortex pattern and their

respective evolution within two different regimes, the zero-field-cooled (ZFC) and the

field-cooled (FC) procedures.

We add the following final introductory remark regarding terminology. By type

I regime, we mean a superconductor in a vortex-free state for every value of the

applied magnetic field until superconductivity is destroyed. We use type II regime to

indicate a superconductor that nucleates one or more vortices before the destruction of

superconductivity, even if the vortex-vortex interaction is not monotonically repulsive.

The outline of this paper is as follows. In Section 2, we present the theoretical

formalism of the 3D Ginzburg-Landau equations and the numerical solution approaches

that guided this work. In Section 3, we determine the critical parameters that define

the crossover from a type I to a type II regime. Next, in Section 4 we demonstrate

several vortex configurations, such as the single-vortex, giant-vortex, and multivortex

states. Finally, we present concluding remarks in Section 5.

2. Theoretical Formalism

The system we investigate consists of a mesoscopic superconducting film of dimensions

(lx, ly, lz) inside a larger box (Lx, Ly, Lz) to take into account the effects of a stray

magnetic field. The superconductor is immersed in a uniformly applied magnetic field

H oriented along the z direction. The geometry of the system is illustrated in Fig. 1. We

consider the dimensions of the domain Ω sufficiently large such that the local magnetic

field equals an externally applied one H at the interface ∂Ω.

Our starting point is the 3D time-dependent Ginzburg-Landau equations (TDGL

for short) that describe the superconducting state. They are coupled partial differential

equations, one for the order parameter ψ and another for the vector potential A, which

is related to the local magnetic field through the expression h = ∇ × A. In order to

fulfil the experimental observation of the dependence of the critical thermodynamic field

Hc(T ) ∝ [1 − (T/Tc)
2], we have modified the GL equations according to [13], so they

now read:

∂ψ

∂t
= − (−i∇−A)2ψ

+
1

(1 + τ 2)2
ψ(1− τ 4 − |ψ|2) , in Ωsc, (1)

∂A

∂t
=

{
Js − κ(0)2∇×∇×A , in Ωsc,

−κ(0)2∇×∇×A , in Ω\Ωsc,
(2)

where Js = Re
[
ψ̄(−i∇−A)ψ

]
is the superconducting current density. In order to

ensure that the local magnetic field h equals the applied field H far away from the

superconducting region Ωsc, we take a sufficiently large domain Ω. We also demand that

no superconducting current flows from the superconductor towards the vacuum, which

means that the superconducting current density vanishes at the superconductor-vacuum



L. R. Cadorim et al. 4

Figure 1. (Color online) Schematic view of the studied system.

boundary ∂Ωsc. Defining a unit vector normal to the superconductor-vacuum interface

n, this asserts that our equations 1 and 2 satisfy the following boundary conditions:

n · (i∇ + A)ψ = 0 , in ∂Ωsc, (3)

∇ ×A = H , in ∂Ω. (4)

Here, the order parameter is in units of ψ∞(0) =
√
−α(0)/β(0) (the order parameter

at the Meissner state), where α(τ) and β(τ) are two phenomenological constants,

τ ≡ T/Tc is the temperature T in units of the critical temperature Tc, lengths

are in units of the coherence length ξ(0), time is in units of the Ginzburg-Landau

characteristic time t0 = πh̄/8KBTc, and the vector potential A is in units of ξ(0)Hc2(0),

where Hc2(τ) is the bulk upper critical field. In order to adjust the phenomenological

constant to the thermodynamic critical field for type I superconductors, Hc(τ) =

Hc(0)(1 − τ 2), the Ginzburg-Landau parameter becomes temperature dependent; we

have κ(τ) = ξ(τ)/λ(τ) = κ(0)/(1 + τ 2), where λ(τ) = λ(0)/
√

1− τ 4 and ξ(τ) =

ξ(0)
√

(1 + τ 2)/(1− τ 2). The TDGL equations are used only as a relaxation method

to achieve a stationary state. They were numerically solved by using the link-variable

method (see, for instance, Refs. [14] and [15]), which was implemented in a GPU-

accelerated forward-time-central-space scheme by using a mesh grid with no fewer than

five points per λ(τ) and ten points per ξ(τ).



L. R. Cadorim et al. 5

3. Critical Parameters

We studied mesoscopic superconducting thin films with a cross section of dimensions

12ξ(0) × 12ξ(0) and variable thicknesses. In the numerical simulations, we used a box

large enough to ensure that demagnetizing effects far away from the superconducting

sample would become small. In all cases studied, the superconducting sample was

initially set to be in the Meissner state and the applied magnetic field adiabatically

increased in steps of ∆H = 10−3Hc2(0) until superconductivity was destroyed (ZFC).

For each value of the film thickness lz, we varied the value of the Ginzburg-Landau

parameter κ(0) until we obtained the crossover between type I and type II behaviours

for that specific d ≡ lz/ξ(0) value.

Figure 2. (Color online) Numerically obtained points and adjusted curves for the

critical values of κ(0) at three different temperatures for a superconductor with cross

section 12ξ(0)× 12ξ(0)

Fig. 2 shows the expressions for the critical values of κ(0), above which vortices are

first seen, for three different temperatures τ = 0.00, τ = 0.75 and τ = 0.85. Below this

curve, the superconductor is type I; above the curve, it becomes type II. By fitting these

curves with a function of the general form κc = Adδ, we were able to find the values of



L. R. Cadorim et al. 6

the A and δ critical parameters. The respective forms of each curve are given by (a)

κc = 0.183d0.464, for τ = 0.00, (b) κc = 0.317d0.438, for τ = 0.75, and (c) κc = 0.351d0.416,

for τ = 0.85.

Contrary to previous studies concerning these critical parameters, we let the

temperature remain entirely explicit so that we could determine the effect of temperature

on the crossover from a type I to a type II regime.

As Fig. 2 shows, for a given value of the thickness d, the critical value of κ(0)

increases as the temperature rises. This behaviour can be explained by the fact that

the transition from type I behaviour to the mixed state happens in an intertype

phase [11, 16, 17], which, unlike both type I and conventional type II, has a non-

monotonical vortex-vortex interaction. This transition between type I and intertype

superconductivity happens for larger values of κ(0) as the temperature approaches Tc
[16], explaining our κc dependence on temperature.

To investigate the influence of the superconductor dimensions on the critical values,

the same procedure was carried out on a sample of a cross section with dimensions

20ξ(0) × 20ξ(0) and a variety of thicknesses. The temperature was set to 0.85Tc. The

resulting curve, presented in Fig. 3, was fit in the same way as the previous ones and

was found to be κc = 0.321d0.487.

As Fig. 3 shows, the critical κc for a superconductor of dimensions 20ξ(0)×20ξ(0) is

below the analogous curve for the superconductor of size 12ξ(0)×12ξ(0). This behaviour

could be explained by a competing effect between the vortex-vortex interactions and

the vortex repulsion from the border of the sample directed towards the centre of the

superconductor. This argument runs as follows. For a fixed value of κ(0), the larger

system presents a greater critical thickness than the smaller one. In these cases, the

vortex-vortex interaction, although it may already be repulsive, cannot overcome the

vortex exclusion from the barrier, which causes a considerable number of fluxoids to

nucleate together at the centre in classic type I behaviour [5]. When the thickness

of the sample is decreased to below dc, the vortex-vortex repulsion due to the stray

magnetic field is now able to suppress the border expulsion, nucleating a small number

of fluxoids, which is a type II behaviour. If we carry out the same analysis on the

smaller sample, we see that, due to their greater proximity, the vortices experience a

more significant repulsion from the border, with a smaller thickness than the previous

case being necessary for the vortex-vortex repulsion to overcome the border effect.

To corroborate this analysis, we studied the current distribution for both values of

the sample dimension for a fixed κ(0) and two thickness values, as shown in the red

points of Fig. 3. In order to compare the two samples on an equal footing, we analysed

their current distributions for values of the applied magnetic field, as shown by the red

points in panel (a) of Fig. 4, for which the penetrated flux was the same for both samples.

In panel (b), we show the intensity of the current at the border of the superconductor

for both dimensions of the sample, with κ(0) = 0.32 and d = 0.9, in such a way that

the smaller sample presents type I behaviour, while the larger one behaves as type II,



L. R. Cadorim et al. 7

Figure 3. (Color online) Numerically obtained points and adjusted curves for the

critical value of κ(0) for the superconductor with two different cross sections. The

temperature was fixed at T = 0.85Tc. The two points indicated in red are analysed in

the text.

as shown in the magnetization curves in panel (a) of Fig. 4. ‡. As the figure shows, the

currents are stronger for the smaller sample, giving support to our previous analysis.

However, this does not mean that the κc(d) curve will be monotonically lower for larger

dimensions; at some point, the curve could go up, approaching the analytical predictions

for infinite thin films [4, 5].

4. Vortex Configurations

We found the vortex configurations for both the ZFC and FC procedures; below, we

present the results for both. For this, we fixed the temperature and took two different

dimensions of the film.

‡ In panel (a) of Fig. 4, the jumps in the magnetization curve signal the entrance of one vortex or

multiple vortices into the sample. The fact that the blue curve presents a single jump indicates that

the system goes directly to the normal state, a type I behaviour, while the two jumps in the red

curve indicate the presence of a vortex state before the transition to the normal state, a classic type II

behaviour



L. R. Cadorim et al. 8

Figure 4. (Color online) (a) Magnetization curve for a sample with dimensions

lx = ly = 12ξ(0) (blue line) and lx = ly = 20ξ(0) (red line). The red points

indicate the values of the applied magnetic field for which both samples present equal

magnetization. (b) The intensity of the currents on the border of the superconductor

for a sample with dimensions lx = ly = 12ξ(0) (blue line) and lx = ly = 20ξ(0) (red

line).



L. R. Cadorim et al. 9

4.1. Zero-Field-Cooled

For the case κ(0) = 0.40 and different film thickness values, we followed the evolution

of the vortex configurations as the applied magnetic field was increased from H =

0 in the Meissner state to the normal state (ZFC). For a system of dimensions

12ξ(0) × 12ξ(0) × 0.4ξ(0) and τ = 0.85, a single vortex first nucleates and reaches

equilibrium at the centre of the sample (see panel (a) of Fig. 5 for the magnitude of

ψ). When the second vortex enters the superconductor, it collapses, with the existing

one forming a giant vortex with vorticity L = 2, still at the centre of the sample (panel

(b)). The same process recurs when the third vortex nucleates, now forming a giant

vortex with vorticity L = 3 at the centre (panel (c)). As the fourth vortex enters,

the existing giant vortex is broken apart, giving way to a configuration of four single

vortices positioned at the vertices of a square around the centre of the sample (panel

(d)). As the magnetic field gets even more intense, superconductivity is destroyed and

the magnetic field is uniform everywhere.

For larger values of thickness d, at least up to d = 0.60, the system moves from

the Meissner state directly to a giant vortex state with triple vorticity, after which it

evolves in the same manner described above. We proceeded in the same manner for

other values of κ(0); similar configurations were obtained. All have a characteristic in

common: namely, the maximum vorticity of the giant vortex state is L = 3. In addition,

the first configuration just after the Meissner state consists of either a single or giant

vortex.

The local magnetic field profiles corresponding to Fig. 5 are presented in Fig. 6,

which shows the intensity of hz in the xy plane. Our results exhibit a new scenario

concerning the giant vortex state in type I superconducting films. It is well known that,

at the centre of a vortex, the field has a local maximum, even for a giant vortex (see, for

instance, Ref. [18]). As panel (a) shows, the field of a single vortex follows this general

property. However, for the L = 2 giant vortex state (panel (b)), hz has a local minimum

at the centre. The flux is spread around the giant vortex core in the form of a ring. This

is even more pronounced for the L = 3 giant vortex shown in panel (c). For the L = 4

single vortex state in panel (d), the magnetic field distribution shows a common pattern,

exhibiting four local maxima. As H increases, the field becomes uniform everywhere.

In order to better visualize the evolution from the single vortex state (Fig. 6

panel (a)) to the giant vortex state with L = 2 (Fig. 6 panel (b)), we followed the

transformation of the local magnetic field profile as the second vortex began to nucleate

at the sample (see Fig. 7 panel (a)) until the two vortices coalesced at the centre of the

sample at the equilibrium state. Fig. 7 shows some metastable vortex configurations

for the applied magnetic field of H = 0.247Hc2(0). Panels (b)-(d) show that, as

the nucleating vortex moves towards the centre of the sample, both vortices suffer a

deformation, combining themselves in a ring-like flux pattern (see panel (f)). Note the

slight asymmetry in this last panel, which is due to the fact that it is still a metastable

configuration but close to the stationary state of panel (b) in Fig. 6.



L. R. Cadorim et al. 10
!

Figure 5. (Color online) Magnitude of the order parameter for four different values

of applied magnetic field in the case of a superconducting thin film of dimensions

12ξ(0)× 12ξ(0)× 0.4ξ(0), κ(0) = 0.40 and T = 0.85Tc.

For larger thicknesses up to dc ∼ 1.4 we found only the L = 4 state of individual

vortices similar to panel (d) of Fig. 6. Above this value of d, the system goes to a

type I regime (see Fig. 2). By studying the dependence of the penetration field Hp on

temperature, we found that, in accordance with previous experimental results [19], the

field at which a vortex is first nucleated increases with decreasing temperature.



L. R. Cadorim et al. 11

Figure 6. (Color online) Intensity of the z component of the local magnetic field for

four values of the externally applied magnetic field for the same system described in

Fig. 5.

4.2. Field-Cooled

We also investigated the vortex state for films of larger dimensions and found the vortex

configurations for both the ZFC and the FC. We learned from this study that the

FC approach produces many more configurations than the ZFC one. Systems that

exhibit type I behaviour in the ZFC method present vortex formation even for smaller

thicknesses in the FC context. Below, we show only states produced by the FC branch.

In Fig. 8 we show four vortex states for a film of dimensions 20ξ(0)× 20ξ(0)× 0.7ξ(0)

and temperature τ = 0.85. For high magnetic fields, the vorticity is large. Thus, the

vortices are so dense that it is necessary to take the magnitude of ψ at a logarithmic

scale to visualize the vortex cores. As we can see, vorticity decays with decreasing

the applied field, with the vortex configuration presenting giant vortices in some cases

(panels (b), (d), and (f)). We note that, even with a low magnetic field applied, when the

shielding currents are small, the vortices do not spread over the sample, which suggests a

non-monotonic vortex-vortex interaction involving short-range repulsion and long-range



L. R. Cadorim et al. 12
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(a)                                        (b)                                        (c) 
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(d)                                        (e)                                        (f) 

Figure 7. (Color online) Metastable vortex states for the applied field H =

0.247Hc2(0). Panels (a) to (f) present the evolution of the local magnetic field profile

from the nucleation of the second vortex to the formation of the giant vortex in a

ring-like pattern.

attraction. The influence of several parameters makes it difficult to fully describe the

vortices’ interactions, but our results do make clear that they are not always repulsive,

as in a conventional type II superconductor.

5. Conclusions

In summary, by numerically solving the 3D TDGL equations for the case of mesoscopic

thin-film superconductors, we were able to obtain the relation between the thickness d

of the film and the κc below which the system presents no vortex formation, behaving

as a type I superconductor. We found that the κc for a given thickness increases with

temperature. We also studied the effect of the sample dimension on the κc curve, finding

that an increase in the dimensions perpendicular to the applied field, with the sample

still being mesoscopic, leads to a decrease in the analysed curve, which is related to the

current distribution over the sample.

Finally, the vortex configurations were studied for several parameters, using two

different procedures. In the ZFC procedure, we followed the evolution of the vortex

patterns with an increasing magnetic field applied. In the FC procedure, we studied

the evolution of the vortex patterns with a decreasing magnetic field applied. In both



L. R. Cadorim et al. 13

cases, we found a novel giant vortex configuration that presents a local minimum of the

magnetic field at the centre of the vortex.

Acknowledgments

The authors thank the following Brazilian agencies for financial support: the São

Paulo Research Foundation (FAPESP) (grant number 12/04388-0), the Coordenação

de Aperfeiçoamento de Pessoal de Nı́vel Superior - Brasil (CAPES) - Finance Code 001,

and the National Council of Scientific and Technological Development (CNPq, grant

302564/2018-7). LRC thanks VUNESP for financial support in the form of a scholar-

ship.

Contributions: ES conceived the work; LRC and TC carried out the theoretical

analysis and the numerical simulations; LRC, ES, and RZ wrote the manuscript, with

contributions from TC.

References

[1] Tinkham M 1963 Physical Review 129 2413

[2] Pearl J 1964 Applied Physics Letters 5 65–66

[3] Abrikosov A A 1957 Soviet Physics JETP 5 1174

[4] Maki K 1965 Annals of Physics 34 363–376

[5] Lasher G 1967 Physical Review 154 345

[6] Callaway D J E 1992 Annals of Physics 213 166–181

[7] Schweigert V A, Peeters F M and Deo P S 1998 Physical Review Letters 81 2783

[8] Shi L M, Zhang L F, Meng H, Zhao H W, Zha G Q and Zhou S P 2009 Physical Review B 79

184518

[9] Berdiyorov G R, Milošević M V and Peeters F M 2006 Physical Review Letters 96 207001

[10] Palonen H, Jäykkä J and Paturi P 2013 Journal of Physics: Condensed Matter 25 385702

[11] Córdoba-Camacho W Y, da Silva R M, Vagov A, Shanenko A A and Aguiar J A 2016 Physical

Review B 94 054511

[12] Dantas D S 2017 Emergent vortex behavior in superconductors and superfluids with single and

multicomponent quantum condensates PhD dissertation Universidade Federal do Ceará, Physics

Department, Fortaleza, Brazil

[13] Ginzburg V I 1956 Soviet Physics JETP 3 621

[14] Gropp W D, Kaper H G, Leaf G K, Levine D M, Palumbo M and Vinokur

V M 1996 Journal of Computational Physics 123 254 – 266 ISSN 0021-9991 URL

http://www.sciencedirect.com/science/article/pii/S0021999196900224

[15] Barba-Ortega J, Sardella E and Aguiar J A 2015 Physics Letters A 379 732 – 737 ISSN 0375-9601

URL http://www.sciencedirect.com/science/article/pii/S0375960114012705

[16] Wolf S, Vagov A, Shanenko A A, Axt V M and Aguiar J A 2017 Physical Review B 96 144515

[17] Vagov A, Shanenko A A, Milošević M V, Axt V M, Vinokur V M, Aguiar J A and Peeters F M

2016 Physical Review B 93 174503

[18] Baelus B J and Peeters F M 2002 Phys. Rev. B 65(10) 104515 URL

https://link.aps.org/doi/10.1103/PhysRevB.65.104515

[19] Ge J, Gutierrez J, Cuppens J and Moshchalkov V V 2013 Phys. Rev. B. 88 174503



L. R. Cadorim et al. 14

!

Figure 8. (Color online) Natural logarithm of the magnitude of the order parameter

for four different values of applied magnetic field for the case of a superconducting thin

film of dimensions 20ξ(0) × 20ξ(0) × 0.7ξ(0), κ(0) = 0.32 and T = 0.85Tc. (a)L = 12

single vortices; (b)L = 10 (eight single vortices and one giant vortex); (c)L = 8 single

vortices; (d) L = 6 (four single vortices and one giant vortex); (e) L = 4 single vortices;

(f) L = 2 (one giant vortex)

.