PHYSICAL REVIEW B 103, 014504 (2021)

Intermediate type-I superconductors in the mesoscopic scale

Leonardo R. Cadorim ,1 Antonio R. de C. Romaguera ,2 Isaías G. de Oliveira ,3 Rodolpho R. Gomes,4

Mauro M. Doria,5 and Edson Sardella 1,*

1Departamento de Física, Faculdade de Ciências, Universidade Estadual Paulista (UNESP), Caixa Postal 473, 17033-360 Bauru-SP, Brazil
2Departamento de Física, Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil

3Departamento de Física, Universidade Federal Rural do Rio de Janeiro, CEP 23890-000 Seropdica, RJ, Brazil
4Instituto de Química, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil

5Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil
and Instituto de Física “Gleb Wataghin,” Universidade Estadual de Campinas, 13083-970 Campinas, São Paulo, Brazil

(Received 30 May 2020; revised 11 November 2020; accepted 16 December 2020; published 7 January 2021)

M. Tinkham [Introduction to Superconductivity, 2nd ed. (Dover, Mineola, NY, 2004), Chap. 4, pp. 135–138]
and P. G. de Gennes [Superconductivity of Metals and Alloys (Benjamin, New York, 1966), Chap. 6, pp. 199–201]
described the existence of an intermediate type-I superconductor as a consequence of an external surface that
affects the well-known classification of superconductors into type I and II. Here we consider the mesoscopic
superconductor where the volume-to-area ratio is small and the effects of the external surface are enhanced. By
means of the standard Ginzburg-Landau theory, the Tinkham–de Gennes scenario is extended to the mesoscopic
type-I superconductor. We find additional features of the transition at the passage from the genuine to the
intermediate type I. The latter has two distinct transitions, namely from a paramagnetic to diamagnetic response
in descending field, and a quasi-type-II behavior as the critical coupling 1/

√
2 is approached in ascending field.

The intermediate type-I phase proposed here, and its corresponding transitions, reflect intrinsic features of the
superconductor and not its geometrical properties.

DOI: 10.1103/PhysRevB.103.014504

I. INTRODUCTION

The classification of superconductors on the basis of
their magnetic properties has been a hard earned knowledge.
Nearly 25 years after the discovery of superconductivity by
Onnes, the experimental measurements of Shubnikov showed
that the magnetic properties of alloys were very different
from those of pure metals [1–3]. The explanation had to wait
another 20 years until the development of the theoretical work
of Abrikosov [4] based on a new and at the time unknown
phenomenological theory, namely the Ginzburg-Landau (GL)
theory. Nowadays, the GL theory enjoys enormous recogni-
tion for its applications in various fields, ranging from phase
transitions to particle theory, since the Higgs model may be
regarded as a relativistic generalization of the GL theory [5].
The classification of superconductors is straightforwardly ob-
tained from the ratio between two fundamental measurable
lengths, namely the London penetration length (λ) and the
coherence length (ξ ). Abrikosov found that the single cou-
pling of the GL theory, κ ≡ λ/ξ , splits the superconductors
into two classes, namely type I and type II, and the critical
value separating them is κ = 1/

√
2. Although his simplified

geometry is beyond reality since there are no boundaries, this
choice is useful in the sense that it excludes any geometrical
factor from entering the classification scheme. This critical

*Author to whom all correspondence should be addressed:
edson.sardella@unesp.br

coupling was also found by Bogomolny [6,7] in the context
of string theory, thus rendering the transitions in κ obtained
here of possible interest to other areas of physics besides
superconductivity. The magnetic difference between type I
and II stems from the existence of a vortex state in type II
that disappears when the normal state sets in at the upper
critical field Hc2. Type I simply does not sustain a vortex
state and goes to the normal state at the thermodynamic field
Hc, where the normal and the superconducting Gibbs’ free
energies become equal. Superconductivity was discovered in
the pure elements, known to be type I with the exception of
Nb, V, and Tc, which are type II. Distinctively, from Shub-
nikov’s time until now, superconductivity has been discovered
in alloys and other composite materials [8], which are mostly
type II. The many new families of high-Tc superconductors
[9] fall in the latter case, such as the cuprates, fullerenes,
MgB2, pnictides, and many others. Nevertheless, unexpected
type-I superconductivity has been found in some alloys, such
as TaSi2 [10], the heavily boron-doped silicon carbide [11],
YbSb2 [12], and more recently in the ternary intermetallics
YNiSi3 and LuNiSi3 [13], rendering them of great interest and
worthy of study. The coupling κ varies greatly among super-
conductors ranging from very high, for the high-Tc materials
YBaCuO7−δ (κ = 95), to very low values, for the pure metals
[9] [Al (0.03), In (0.11), Cd (0.14), Sn (0.23), and Ta (0.38)]
and also for some of the alloys [12,13], such as YbSb2 (0.05)
and YNiSi3 (0.1). Since ξ decreases with disorder, doping the
material by impurities can produce adjustable κ compounds
that allow for the study of transitions in this coupling.

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https://doi.org/10.1103/PhysRevB.103.014504


LEONARDO R. CADORIM et al. PHYSICAL REVIEW B 103, 014504 (2021)

According to Tinkham [1] and de Gennes [2], surface
states split type-I superconductors into genuine and interme-
diate classes that comprise the coupling ranges κ < κc and
κc < κ < 1/

√
2, respectively, where κc = 0.417. Although

this transition is driven by surface effects, it solely reflects
an intrinsic property of the superconductor, namely κ , as
discussed below. There has been an intense search for this
genuine-intermediate transition both theoretically [14,15] and
experimentally [16–19] in the 1970s, but after this period
it became an elusive topic. The existence of the transition
was even questioned [20] and its study no longer pursued.
Now that type-I alloys were found, the classification of su-
perconductors has acquired a renewed interest. In this paper,
we pursue this study in the mesoscopic scale, which offers a
unique framework since surface effects are enhanced there. In
the bulk (no surface) the relation Hc2 = √

2κHc elucidates the
difference between types I and II since for κ < 1/

√
2 (κ >

1/
√

2), Hc2 < Hc (Hc2 > Hc), thus a type-I (type-II) super-
conductor. Interestingly, Tinkham has stressed in his book [1]
that both fields Hc2 and Hc are directly measurable in type-I
superconductors. The normal state is retained below the ther-
modynamic field Hc until the lower field Hc2 is reached [17],
where the order parameter (magnetization) abruptly becomes
nonzero. From the other side evolving from the superconduct-
ing state, this lasts beyond Hc2 until Hc is reached and the
normal state is recovered. However, the presence of a surface
modifies the above bulk analysis since superconductivity is
extended beyond Hc2 to exist with thickness ξ around the
external boundary. Saint-James and de Gennes [21] found the
critical field Hc3 > Hc2 whose value in the case of a flat inter-
face is Hc3 = 1.695Hc2 [21–27]. Hc3 is also present in type-I
superconductors [28] and signals several processes, e.g., the
expulsion of magnetic flux [29,30]. In fact, it is Hc3 that gives
rise to the genuine and intermediate type-I superconductors
associated with Hc2 < Hc3 < Hc and Hc2 < Hc < Hc3, respec-
tively, such that κc is obtained from Hc3 = Hc.

In this paper, we report properties of the genuine-
intermediate transition such as its critical κ and also other
transitions in κ in the intermediate phase, as seen by isother-
mal magnetization M(H ) curves. These several transitions
can be experimentally investigated using the ballistic Hall
magnetometry technique [31] applied to submicron-sized su-
perconductors. Type-I mesoscopic superconductors have been
investigated both theoretically [32] and experimentally [33],
however the genuine-intermediate transition in κ is first con-
sidered here and found to acquire new properties. As one goes
from the macroscopic to the mesoscopic scale, κc gives rise to
κc1 and also to the transitions κc2 and κc3 inside the intermedi-
ate phase. In descending field starting from the normal state,
the magnetization can display paramagnetic regions, but it be-
comes diamagnetic at any applied field provided that κ < κc2.
Hereafter, we call this the dia-para transition. In ascending
field, the magnetization (−M) has a nearly linear growth
(Meissner state) up to a maximum and next undergoes an
abrupt fall, and a residual magnetization regime is reached that
only exists if κ > κc3. The Meissner state is followed by the
disappearance of the magnetization for κ < κc3. This residual
magnetization signals the quasi-type-II class and is caused
by giant vortices (for a discussion about giant vortices in
mesoscopic superconductors, see, for instance, Refs. [34,35]).

We remark on the presence of several notable fields in our
study. In the up branch there are H ′

c (the peak of −M) and H ′′
c

(the vanishing of the magnetization). They are very near to
each other for κ < κc3 but not for κ > κc3. Both critical fields
fall above Hc (see the Supplemental Material), and so they
are inside a region of metastability since the superconducting
state there has higher Gibbs free energy than the normal state.
In the descending branch we define H ′

c3, where the magne-
tization becomes nonzero and the superconducting state sets
in. As shown here, these fields, as well as the M(H ) curves,
are strongly dependent on κ in the type-I domain. We bring
numerical evidence that the genuine-intermediate mesoscopic
transition takes place at a coupling lower than the macroscopic
one, κc1 < κc. Interestingly the dia-para transition occurs at
κc2 ≈ κc, thus near to the macroscopic transition, which has
been a fortuitous coincidence thus far. The transition to the
quasi-type-II class takes place for κc3 < 1/

√
2.

Our numerical analysis was carried out on a very long
needle with a square cross section of size L2 in the presence
of an applied field parallel to its major axis. The needle is
sufficiently long such that the top and the bottom surfaces
can be ignored and just a transverse two-dimensional cross
section needs to be considered. The square cross section is
the most suited to our numerical procedure, which is done
on a square grid. The boundary conditions are smoothly im-
plemented in this geometry, namely of no current exiting the
superconductor and that at the surface the local field meets the
external applied field. Nevertheless, our major findings hold
independently of the selected cross-section geometry, though
the value of the critical fields and of the delimiting κci may be
affected by it. We look at several cross-section sizes, namely
L = ρλ and ρ = 8, 12, 16, 24, and 32.

It is well known that the GL theory is the leading term of an
order-parameter expansion derived from the microscopic BCS
theory [2,36,37]. If the next-to-leading-order corrections are
included [37], an intermediate phase emerges in the diagram
κ versus T in between the type-I and -II domains. However,
the analysis of Vagov et al. [36,37] does not take into account
surface effects, whereas here the intermediate phase is solely
due to these surface effects [38], and for this reason the in-
termediate phase is found already in the standard GL theory
level. The choice of an infinitely long system, instead of being
a limiting factor, really expands the importance of the present
results once it allows us to see intrinsic effects. Geometrical
factors [39–42] hinder the observation of the intrinsic transi-
tions observed here. It is well known that a sufficiently thin
type-I superconductor turns into a type-II one by a change of
its thickness. The geometry of the cross section affects the
κ values where transitions take place but not their existence,
which only reflects the ordering among the critical fields.

II. THEORETICAL FORMALISM

The basis of our dimensionless treatment of the GL theory
is λ and Hc2 = �0/2πξ 2, which renders the free energy,

G =
∫ [∣∣∣∣

(
− i

κ
∇ − A

)
ψ

∣∣∣∣
2

− |ψ |2 + 1

2
|ψ |4

]
d3r

+
∫

(h − H)2 d3r, (1)

014504-2



INTERMEDIATE TYPE-I SUPERCONDUCTORS IN THE … PHYSICAL REVIEW B 103, 014504 (2021)

in reduced units. Lengths are in units of λ; the order parameter
ψ is in units of ψ∞ = √

α/β, where α and β are the two
phenomenological constants of the GL theory; magnetic fields
are in units of

√
2Hc; and the vector potential A is in units of√

2λHc. The GL equations become

−
(

− i

κ
∇ − A

)2

ψ + ψ
(
1 − |ψ |2) = 0, (2)

∇ × h = Js, (3)

where Js = IR[ψ̄ (− i
κ
∇ − A)ψ] is the superconducting cur-

rent density.
The GL equations were solved numerically within a suit-

able relaxation method and upon using the link-variable
method as presented in Ref. [43]. For this, we used a mesh-
grid size with �x = �y = 0.2λ. The applied magnetic field is
adiabatically increased in steps of �H = 10−3

√
2Hc for both

up and down branches of the field. In each simulation, κ was
held fixed and the stationary state at H ∓ �H was used as the
initial state for H for up and down cycles, respectively.

The emergence of superconductivity in a long mesoscopic
cylindrical (R ∼ λ) in the presence of an applied external
field has been studied by Zharkov et al. [44–47]. Their
approach is limited to the search of solutions of the Ginzburg-
Landau differential equations with radial symmetry, which
limits the search to central vortex states. In our numerical
search through the link variable method, we find the presence
of point vortices forming various geometrical patterns inside
the superconductor that fall beyond their description. Hence
the observation of the present κci transitions is beyond the
scope of their framework since they are limited to a subset
of the possible vortex states.

III. RESULTS AND DISCUSSION

The intermediate and the genuine type-I classes are dis-
tinguishable by their magnetic properties. In decreasing field,
the genuine class features a direct and abrupt change from
the normal state to the Meissner state, i.e., vortices are never
trapped inside the superconductor. In the same situation, the
intermediate class displays vortices, which are trapped inside
either as single or giant ones and then are gradually or sud-
denly expelled. The two classes are associated with specific
κ ranges, and to determine them we have performed a series
of numerical simulations varying κ in steps of �κ = 0.01.
Within this precision, we were able to numerically obtain κc1,
κc2, and κc3 for all the L’s under investigation. In what follows,
we report properties of the genuine-intermediate, dia-para,
and quasi-type-II transitions, the latter two being inside the
intermediate type-I class.

Figure 1 features the genuine-intermediate transition
through the number of vortices, N , trapped at H ′

c3. The su-
perconductor response is markedly distinct according to κ ,
and this is exemplified here for L = 16λ and 24λ. Above κc1,
which is equal to 0.28 for L = 16λ, and 0.19 for L = 24λ, N
varies according to κ , thus corresponding to the intermediate
type-I class. However, below κc1, N drops to zero showing
that no vortex enters the needle at H ′

c3, which characterizes
the genuine type-I class. The insets of Fig. 1 depict the mag-
netization curve of two selected κ values belonging to the two

FIG. 1. The vorticity N at H ′
c3, the field where the supercon-

ducting state sets in decreasing field, is plotted as a function of κ .
The transition κc1 corresponds to N = 0 marking the onset of the
genuine type-I class and the end of the intermediate type-I. The
insets show typical magnetization curves for the up (blue) and down
(red) branches, above and below this transition. The black ellipse
highlights the spike state, which is the last possible vortex state in
the intermediate type-I class. In the spike state, a vortex nucleates
and is immediately spelled from the superconductor.

classes, chosen as 0.28 and 0.33 for L = 16λ. The left inset
depicts a magnetization that goes directly from the normal
(M = 0) to the Meissner state, while the right inset shows a
spike highlighted by the black ellipse that corresponds to the
coalescence of flux in the form of vortices and their subse-
quent exit at an infinitesimally lower field. The magnetization
curves show the up (blue line) and down (red line) branches
for both insets.

Figure 2 features the dia-para transition. The main panel
presents κc2 as a function of L. The small red circles indicate
the numerically obtained κc2 values of 0.4425, 0.4175, 0.4025,
0.4025, 0.4075, 0.415, and 0.415 for L/λ equal to 8, 9, 10, 12,
16, 24, and 32, respectively. Although κc2 is a nonmonotonic

FIG. 2. The magnetization in descending field reveals the tran-
sition at κc2, shown in the main panel as a function of L, from
paramagnetic to diamagnetic response. Insets (a) and (b) display
typical magnetization curves below and above the transition. Inset
(b) shows a still paramagnetic magnetization, while in panel (a) it
has become totally diamagnetic.

014504-3



LEONARDO R. CADORIM et al. PHYSICAL REVIEW B 103, 014504 (2021)

FIG. 3. The magnetization is a maximum at the field H ′
c (the

peak of −M) and vanishes at the field H ′′
c where superconductivity is

destroyed and the normal state is restored. H ′′
c vs κ is shown here and

clearly points to a κc3 transition. The insets show typical magnetiza-
tion curves below and above this transition. The left inset shows the
situation that H ′′

c ≈ H ′
c and no vortices are present, whereas the right

inset illustrates the situation H ′′
c > H ′

c in which vortices do nucleate
in the region highlighted by an ellipse.

function of L, it asymptotically approaches a limiting value
for large L, suggestively close to κc, which is the value that de-
limits the genuine-intermediate transition in the macroscopic
limit. Insets (a) and (b) of Fig. 2 depict the transition occurring
at κc2 for the case of L = 24λ through two selected values
of κ , each characterizing one side of the transition. Inset (a)
shows the typical diamagnetic behavior for κ = 0.4 < κc2,
and the magnetization is always negative for any value of the
applied field. Inset (b) presents the magnetization for the case
with κ = 0.43 > κc2 at which paramagnetic regions exist in
the down branch, and it alternates with diamagnetic regions,
thus not qualifying as a totally diamagnetic response.

Figure 3 shows the quasi-type-II transition, signaled in the
ascending magnetization by two notable fields, namely H ′′

c
and H ′

c. For κ < κc3, the maximum of the magnetization is
immediately followed by its sudden drop to zero, H ′′

c ≈ H ′
c.

However, for κ > κc3 the two fields depart from each other,
and for increasing κ , H ′′

c − H ′
c also increases and vortices are

observed in the superconductor. The kink in the curve H ′′
c ver-

sus κ shown in Fig. 3 defines κc3. The insets of Fig. 3 display
situations below (left) and above (right) the transition. The left
one shows the magnetization curve going from the Meissner
state directly to the normal state, whereas the right one, in
contrast, presents a vortex state in between the Meissner and
normal states. The black ellipse in this inset highlights the
vortex state region. Remarkably, this transition is found to oc-
cur at κc3 = 0.54 for any size L. Interestingly, it is found that,
apart from small numerical deviations, H ′′

c ≈ H ′
c3 for κ > κc3.

Concerning the Gibbs free energy of the quasi-type-II class,
it is negative though very close to zero. This small negative
value is still sufficient to render it slightly below the normal
state energy, which is zero (see the Supplemental Material
[48]). This vortex regime is subsequent to the peak of the mag-
netization, which lies in an energetically metastable regime.
This is in contrast with the standard type-II superconductor,
where the peak of the magnetization is within a totally stable

FIG. 4. κ-L phase diagram for the system under study in the
ranges 0.125 < κ < 0.8 and 8λ < L < 32λ. As given in the figure,
the dark blue, the light blue, the green, and the yellow regions cor-
respond to genuine type-I, intermediate type-I with diamagnetism,
intermediate type-I with the occurrence of paramagnetism in the
down branch, and quasi-type-II, respectively.

regime. We find that κ = 0.8 is still quasi-type-II behavior,
but the upper boundary, which ought to be connected to the
stability of the magnetization peak, is not specified.

Figure 4 displays the κ versus L phase diagram containing
all the transitions discussed here. The GL theory ceases to
be valid at L = ξ , and for this reason the diagram features
κ � 0.125, which guarantees L > ξ for all L considered. The
delimiting curves separating any two regions are obtained
by a fitting process. The κci(L) lines are indicated at the
right margin of the figure, and they separate the four regions,
namely genuine and intermediate, the latter split into sub-
classes known as dia, para, and quasi-type-II. Hence Fig. 4
is the generalization for the mesoscopic superconductors of
the de Gennes–Tinkham transition found at κc for the macro-
scopic superconductor.

IV. CONCLUSIONS

In summary, we show that mesoscopic type-I supercon-
ductors have intrinsic transitions in κ . The genuine type-I
behavior is only possible below κc1, and above it, vortices
exist in this so-called intermediate type-I class that has a rich
structure with a transition from paramagnetic to diamagnetic
response, in descending field (κc2), and a quasi type-II behav-
ior, in ascending field (κc3).

ACKNOWLEDGMENTS

L.R.C. thanks the Brazilian Agency Fundação de Amparo
Pesquisa do Estado de São Paulo (FAPESP) for financial
support (Process No. 20/03947-2). A.R.d.C.R. M.M.D., and
E.S. thank FACEPE and CAPES for financial support with
Project No. APQ-0198-1.05/14. E.S. thanks the Brazilan
Agency Fundação de Amparo Pesquisa do Estado de São
Paulo (FAPESP) for financial support (process no. 12/04388-
0). We also gratefully acknowledge the support of the
NVIDIA Corporation with the donation of GPUs for our
research.

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curves, the energy curve for the vortex configuration in the
quasi type-II phase and animations of the order parameter for
different values of κ .

014504-6

https://doi.org/10.1088/1361-648X/ab4a4a
https://doi.org/10.1088/1361-648X/ab5379
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http://link.aps.org/supplemental/10.1103/PhysRevB.103.014504